That brings today's live coverage of the EU migrant crisis to an end. For further updates please check the main BBC News site for the latest information.
- Thousands of migrants crossed into Austria amid jubilation after Hungary removed restrictions on transit
- Many migrants are now continuing their journey by train to Vienna
- Hundreds have now arrived in Munich to be greeted by applauding crowds
- Germany has promised to grant refugee status to those who fled the Syrian conflict
- EU foreign affairs chief Federica Mogherini has warned that the migrant issue is "here to stay"
- Copyright: AP
Meanwhile further south along the route to Germany, more migrants and refugees have arrived from Turkey. This group is waiting to board a bus at the Greek port of Piraeus, which will take them to a metro station.
The BBC's Gavin Lee reports from Hungary that around 1,000 people are marching from Budapest station and are around 16km (10 miles) into the journey.
The Hungarian government is insisting there will be no repeat of the help for migrants trying to reach Austria, saying that buses were provided as a one off because of safety fears.
Our correspondent says policy at Budapest's main train station appears to have changed. Previously those without travel documents were stopped from travelling on international services, but today they were allowed to buy tickets for stations within Hungary, closer to the border.
For those that don't have money for tickets, another march to Austria is being planned for this evening.
One volunteer in Nickelsdorf, Johannes Noebel, said he felt compelled to help.Quote Message: When I saw these pictures of the people walking on the highway yesterday night, I said...: we have to come and welcome them. And so when we heard that they will all end up here, we took a truck and loaded it full with water and food, and came here to help.Quote Message: I think that it's really important to give a warm welcome to the people, also the many, many children that you see here.
An Austrian Red Cross employee in the border town of Nickelsdorf has told the BBC that migrants are being offered temporary shelter in a large tent filled with beds, on the outskirts of the town.
The tent was set up as part of a rock festival which ended a few weeks ago.
Authorities decided to leave it standing in anticipation of migrants arriving from Hungary.
- Copyright: Reuters
Migrants have been arriving at Munich's main train station during the day, where they have been greeted by locals, some bearing gifts of sweets and chocolate.
The BBC's Ben Brown tweets:
A BBC News producer says that members of a group marching from Budapest are using technology to help them reach their destination:
The BBC's Anna Holligan reports from a rest stop some 17km west of Budapest which has been set up by local people for migrants marching to the Austrian border.
The volunteers, some of them British, have rucksacks and waterproofs as well as fruit and bottles of water to distribute to a group of migrants being escorted by police, our correspondent says.
The Guardian has been speaking to some of those in Germany who were waiting at Munich's train station to welcome people to the country.
Hedy Gupta, who handed out chocolate to the new arrivals, told the newspaper: "We just wanted them to know that the torture is over.
"I have children and a five-year-old grandchild of my own and when I think what they have been through, these children, it leaves me on the ground.”
BBC Radio 4's chief correspondent tweets:
BBC News Europe reporter tweets:
Nearly 2,500 people have arrived at the Greek port of Piraeus from the island of Lesbos, Reuters reports.
Latest from Nickelsdorf, Austria: An Austrian police officer tells the BBC's Ben Brown that nearly 5,500 refugees have been transported to Vienna's Westbahnhof after the authorities laid on extra trains and buses.
The policeman said about 200 to 300 refugees are currently indoors in Nickelsdorf and will be brought to the station just before the next train is due to depart, to save them waiting outdoors.
- Copyright: BBC
The BBC's Berlin correspondent Jenny Hill watched as hundreds of migrants, mostly from Syria, arrived at Munich train station to applause after boarding trains in Austria.
A large group of Germans who had gathered to welcome them cheered their arrival, with many handing out sweets and water.
European Union foreign ministers have discussed the possibility of setting up EU centres for refugees outside Europe as a way to deal with the migration crisis, Reuters reports.
Allowing refugees to request asylum in the Middle East and Africa could help stop thousands of people attempting the perilous journey to reach Europe, it says.
But it reports that EU foreign policy chief Federica Mogherini has warned that the idea would require "enormous resources".
"We cannot destabilise countries that are already facing enormous challenges when it comes to camps and refugees in the communities. And this would require enormous resources from our side to set them up," she says.
Sweden is the only European country with a majority favourable towards non-EU immigration, The Independent reports.
It says that even in Germany, where Chancellor Angela Merkel has insisted that the nation is well equipped to meet the refugee crisis, not everything is perfect, because it continues to be plagued by far-right attacks on refugee accommodation.
Citing a recent Eurostat survey, the paper says that between 71 and 77% of Swedes approve of non-EU immigration.
At the bottom of the list come Italy, the Czech Republic, Slovakia, Estonia and Latvia where only between 15 and 21% welcome immigrants.
Germany can cope with a record influx of refugees this year without raising taxes and without jeopardising its balanced budget, Chancellor Angela Merkel says, according to Reuters.
"We cannot just say 'Because we have a difficult task now, the balanced budget or the issue of debt are no longer important'," Mrs Merkel said in her weekly video podcast.
And in an interview with local newspapers, she said taxes would not be raised because of the migrant crisis.
- Copyright: Reuters
"Europe needs to wake up [in relation to the migrant crisis] and the time for reverie is over," Austrian Interior Minister Johanna Mikl-Leitner says.
"Now the continent of Europe is challenged. In this great challenge the entire continent has to give a unified answer.
"Whoever still thinks that withdrawal from the EU or a barbed wire fence around Austria will solve the problem is wrong."
David Simmonds, from the Local Government Association, has told the BBC that the planned arrangement for funding migrants who arrive in Britain for their first year only will "leave local council tax payers picking up the bill" for a decision made by central government.
Council leaders have called on ministers to ensure that resources are in place to support people as they come to Britain.
The UK's Home Office says local authorities will only receive funding for the first year if they accept refugees who have been brought from near Syria under a government scheme.
David Cameron's promise to accept "thousands" of Syrian refugees will be delivered under the government's Vulnerable Persons Relocation scheme.
Under this scheme, central government fully funds the costs of health, education and social care costs as well as the provision of tailored integration support for 12 months.
But officials confirmed that once those 12 months are up, the costs of supporting the refugees will fall to the local authorities where they are living.
Thousands of people are on their way to Germany via Hungary - but that's only part of their journey. This map shows the long route taken by migrants as they head towards northern Europe.
You can read about why the EU is struggling with migrants and asylum here.Copyright: BBC
The usual diplomatic conviviality unravelled at an EU foreign ministers' meeting in Luxembourg on Saturday, Reuters says, as they failed to agree on any practical steps out of the crisis.
They are especially at odds over proposals for country-by-country quotas to take in asylum seekers, it reports.
- Copyright: Reuters
Hungary says that it has recorded some 165,000 migrants entering the country so far this year.
On Saturday at least 500 people took part in another march from Keleti station towards Austria, after about 800 migrants reportedly walked out of Hungary's second largest refugee camp outside the eastern city of Debrecen.
A World Have Your Say on the issue of migration is being broadcast live now on the BBC World Service, and the show is speaking to volunteers in Vienna and policy makers from Europe. Questions are also being taken from listeners.
- Copyright: PA
David Simmonds, from the Local Government Association, says musician Bob Geldof's offer to give a home to Syrian refugees is "pie in the sky".
Mr Geldof said he felt "profound shame" at the growing refugee crisis and has offered to put up four families in his homes in Kent and London.
But Mr Simmonds said: "If Bob Geldof is willing to make that offer I'm sure his local council, which will already have a lot of people on its housing waiting list, will be very happy to bring them around this afternoon."
EU foreign affairs chief Federica Mogherini says that the refugee influx is here to stay and member states must adjust to that new reality.
"The sooner we accept it, the sooner we will be able to respond effectively [and] unite as Europeans," Ms Mogherini is quoted by the AFP news agency as saying, after a two-day informal meeting of EU foreign ministers.
"It affects all of us. A few months ago, it was Italy, Greece and Malta. Now it is Hungary and it could [be the] turn of other member states in the future," she said.
- Copyright: Reu
Ivand, 18, from Kobane, Syria takes a selfie with his friends as they walk along a railway track after crossing into Hungary from the border with Serbia.
Hungary's police chief says that buses are no longer being provided to transfer migrants to the Austrian border, and that the measure that transported thousands overnight and on Saturday morning was a "one-off".
Europe and Britain must offer asylum to those genuinely fleeing persecution but also need to boost aid spending, defeat smuggling gangs and tackle the conflict in Syria to curb the migrant crisis, Britain's Finance Minister George Osborne tells the Reuters news agency.
"It's absolutely clear we need a comprehensive plan across Europe," he said on the sidelines of a meeting of G-20 finance ministers in Turkey.
"Yes, we must offer asylum to those who are genuinely fleeing persecution. Countries like Britain always have, we are one of the founders of the asylum system. We will take, as the [British] prime minister said, thousands more.
"But at the same time, you've got to make sure you've got aid going into the refugee camps on the borders ... We've got to defeat these criminal gangs who trade in human misery and risk people's lives and kill people."
Polish Prime Minister Ewa Kopacz reiterates her country's commitment to accept 2,000 migrants. "We are committed to solidarity but it has to be a responsible solidarity," she is quoted by Reuters as saying.
The influx of refugees from countries such as Syria, Iraq and Afghanistan is bad news for potential migrants to Germany from western Balkan countries such as Serbia and Macedonia, The Guardian reports.
It says that up to 75,000 asylum requests this year by migrants from south-eastern Europe are expected to be rejected by Germany.
The country has deported more than 10,000 foreigners so far this year, many of them from the Balkans, the paper quotes Germany's Der Spiegel magazine as saying.
The BBC's Jenny Hill reports that some Germans are waiting at train stations to distribute toys to refugee children arriving in the country:
- Copyright: Getty Images
A migrant holds up a newspaper with a picture of German Chancellor Angela Merkel after his arrival from Austria at Munich's main railway station.
A refugee who made it across the border from Hungary into Austria has told the BBC's Nick Thorpe of his determination to reach Germany. Sina Omid said that he had been travelling for more than one month, spending more than a week in Hungary.
"If they help us to go in Germany, I am so happy," he said.
The BBC's Mark Lowen in the Turkish coastal town of Bodrum says that despite the dangers, there are still families - including young children - waiting to make the journey to Greece.
Email Message: John Wantage from Oxford emails:
"We have to break the link between trafficking and migration, so we need to get rid of the reward for trafficking. The UK is absolutely right to only take people from the camps. At the same time, we need to remove the link between asylum and migration by removing the right to apply for a passport after five years."
Around 6,500 refugees have reached Austria since neighbouring Hungary began ferrying stranded migrants to the border, Reuters reports Interior Minister Johanna Mikl-Leitner as saying.
She added around 2,200 of them are already on their way to Germany.
BBC Radio 4's chief Today correspondent, Matthew Price, tweets this image of a train at the Austrian/Hungarian border, which has arrived to take migrants to Munich.
However, he points out that the train began its journey in Budapest, from where the migrants themselves marched or were bussed, after becoming frustrated at not being allowed to leave by the authorities. They are now being allowed to get on board, further down the line.
- Copyright: BBC
BBC foreign correspondent Bethany Bell tweets footage of migrants attempting to board a train on the Austrian/Hungarian border, in which male migrants are told to allow women and children aboard first.
Email Message: Rosemary Hill emails:
"Wonderful to see the welcome and help being given in Europe - including by ordinary decent people in Hungary - to those who desperately need it. I hope (but doubt) that Viktor Orban is hanging his head in shame this morning, and likewise the leaders of the Gulf Arab countries whose silence throughout this crisis has been positively deafening. The next time they start bleating that 'the world community' should pull them out of a hole, I for one will be turning an equally deaf ear."
- Copyright: EPA
Hungary will deploy police along its southern border after 15 September to stop an influx of refugees, Prime Minister Viktor Orban is reported as saying on Saturday by Reuters.
The PM will also send in the military if parliament approves a government proposal, the agency added. "The big changes will come after September 15... and we'll bring the border under control step by step," Mr Orban told a news conference.
"We'll send in the police, then, if we get approval from parliament, we'll deploy the military."
- Copyright: AP
In her comments earlier to that news conference, the EU's foreign policy chief, Federica Mogherini, stressed European values. EU countries needed to cast aside their differences and stand together to deal with the problem.
"The time for blame games is over, it's time for taking decisions, turning decisions into actions, and doing it united, as Europeans.
"Only in this way will we have the possibilities to face this issue, this urgency, this dramatic event, keeping faith to our European values."
More details are coming in about the 450 migrants who arrived on a special train at Munich railway station. They will now be escorted onto a city train to take them to an emergency registration centre nearby, police say.
Police have said that authorities expect between 5,000 and 10,000 refugees to come to Germany from Hungary via Austria on Saturday.
Europe's migrant crisis has been top of the agenda at a meeting of European Union foreign ministers in Luxembourg. The EU's foreign policy chief, Federica Mogherini, said the problem was not going to disappear quickly.
"This is not an emergency - it's an urgency we are facing - but it is not something that starts that day and finishes that day.
"It is here to stay, and the sooner we accept it psychologically and politically, the sooner we will be able to respond in an effective way, and to manage in an effective way."
Hundreds of migrants have arrived in Munich station on a train from Austria, reports say.
One male migrant in the border town of Nickelsdorf told APTN he was relieved to have arrived in Austria.
"Now, it's very, very happy. Now we are free. For five days in Hungary, we were in a very, very bad position. And Allah, thanks God, thanks God."
The BBC's Imelda Flattery tweets that Hungarian police await the 1,000 or so migrants who are marching from Budapest, towards Austria.
The UN Refugee Agency has praised Austria and Germany for taking in thousands of migrants who crossed the border from Hungary. It said in a statement on Saturday that this was "political leadership based on humanitarian values".
The Geneva-based body also praised community groups and the public in Austria and Germany for helping provide a welcome to those arriving. The agency says "a remarkable outpouring of public response'' is driving some governments to change their stance on accepting migrants.
But it said "the concentration of refugees and migrants in a small number of countries willing to receive them is not a sustainable solution.''
Email Message: Rosamund Whittam writes:
"I'm just surprised that there has been no mention of Assad on the news for days. What is he doing, how is he reacting to the many people leaving his country and why is there no political pressure on him to prevent the situation that is so dire for his own people?"
- Copyright: Getty
Migrants arriving in Austria from Hungary were pleased to be given food and a warm welcome.
More details on that offer from the Finnish Prime Minister Juha Sipila to make one of his homes available for migrants. It will be made available at the new year, he said.
"I know many others have made a similar decision, people either host refugees in their homes or make [other] space available.
"And I think that people ought to think what each one of us can do. People can do voluntary work, or collect funds or clothes that are needed. I hope for this kind of volunteer spirit."
Austria will not use force to stop thousands of refugees streaming into the country from Hungary, Interior Minister Johanna Mikl-Leitner is quoted by Reuters as saying.
The minister appealed to the rest of Europe to help shoulder the burden of the mass influx.
"More than 3,000 refugees reached Austria last night. Each refugee can of course apply for asylum in Austria and is being informed of this possibility," she said in a statement after a crisis task force convened in Vienna.
The outpouring of emotion over the death of a Syrian toddler has blinded us to the roots of this humanitarian crisis, argues Matthew Paris in The Times.
"The distinction between an 'economic migrant' and a political asylum-seeker is impossible to maintain when entire countries, their economies, their administrations and their law and order break down," he writes.
"Whether that dead child’s father was escaping from political persecution, or simply from chaos and despair, I defy you to adjudicate and I doubt he could."
The first train to arrive from Austria to Germany has arrived in Munich was carrying 167 migrants, according to the Associated Press. They were among a larger group who had traveled by bus from Hungary to Austria.
Federal police spokesman Simon Hegewald told AP a specially chartered train from Salzburg, Austria, with several hundred migrants on board is expected in Munich around noon GMT.
Human rights group Amnesty International has welcomed the initiative to clear Hungary's humanitarian traffic jam.
"After endless examples of shameful treatment by governments of refugees and migrants in Europe, it is a relief to finally see a sliver of humanity. But this is far from over, both in Hungary and in Europe as a whole,'' Amnesty's deputy director for Europe, Gauri van Gulik, is quoted by the AP news agency as saying.
"The pragmatic and humane approach finally applied here should become the rule, not the exception."
- Copyright: BBC
BBC Radio 4's chief correspondent Matthew Price has tweeted footage of children on their way from Hungary to Austria, who he says are "so well behaved. After so much".
Email Message: C Calver emails:
"I sympathise with the eastern Europeans. They have the right idea. Close the borders. Letting these crowds of who know what just push their way in is bizarre. These “migrants” have treated Hungary with contempt forcing their way into a country and refusing to recognise the procedures. When they passed through Turkey, which is a safe place they ceased to be refugees and became economic migrants? “Germany! Germany! Germany!” they chant. What more proof do u want? The want the good life rather than safety.
- Copyright: EPA
Many migrants are beginning to arrive at Munich train station on Saturday morning. Germany is expecting 10,000 arrivals out of Hungary, now people there have been allowed to travel onwards.
British MPs will have a full day’s debate on the refugee crisis following Prime Minister’s Questions on Wednesday, the Guardian reports. It says that the Scottish National Party (SNP) has decided to use its opposition day to raise the issue.
The SNP will lead the debate by calling for urgent action from the British government, which it argues has "failed to show moral or political leadership" over the migrant issue.
Europe's treatment of migrants evokes memories of the the continent's darkest hour, The New York Times says.
“It was horrifying when I saw those images of police putting numbers on people’s arms,” the paper quotes Hungarian Chief Rabbi Robert Frolich as saying.
“It reminded me of Auschwitz."
The human tide south of Hungary is rising, officials quoted by the AP news agency have warned, as more westward-bound travelers arrived in Budapest within hours of the mass evacuation of the capital's central rail station.
- Copyright: EPA
A steady stream of migrants have been crossing from Hungary in Austria throughout Saturday morning.
Hungarian authorities say there will be no further buses or trains to take people to the Austrian border. It says it acted out of fear for the safety of those who had been walking towards the border, and is now reviewing the situation.
A train from Austria carrying several hundred refugees is expected to arrive in Munich around noon GMT, Reuters reports.
- Copyright: BBC
The Malta-based search and rescue charity Migrant Offshore Aid Station (Moas) says it has received more than €1m (£730,000, $1.1m) in donations from around the world in the past 48 hours, since the outcry over the images of three-year-old Alan Kurdi whose body washed up on a Turkish beach.
Moas has a ship, carrying drones and medical supplies, which helps rescue people attempting to cross the Mediterranean Sea.
The BBC's Imelda Flattery reports that more than a thousand migrants have started walking from Budapest station to Austria, a march that has occurred for the second day running.
The BBC's Imelda Flattery reports that many more migrants are arriving at Keleti station in Budapest, Hungary.
Germany is prepared to receive 10,000 migrants over the next few hours, Reuters reports, with Bavarian state police saying they expect the first refugees to arrive in Germany around midday GMT.
Hungarian government spokesman Zoltan Kovacs says his country has fulfilled its obligations with regards to migrants.
"Obviously what we promised, we are fulfilling, that is we are keeping law and order, not only at the borders... but also on the roads and railways of the country, as well as the Hungarian side of the border."
British councillor David Simmonds, from the Local Government Association, says councils in the UK must have the resources to deal with any new influx of migrants.
"It costs around £50,000 a year to support a child in the care system. It costs around £30,000 to provide a school place.
"And with the pressure we already have on UK public services, the key question is making sure the resources are there to ensure that communities that are receiving new arrivals recognise that's not an unfair burden on them."
The Hungarian government had been criticised for preventing thousands of people, many of them Syrian refugees, from leaving the country.
On Friday night, the Hungarian authorities changed tack and laid on buses to take people to the Austrian border.
Throughout the night, the BBC's Matthew Price has been travelling with some of the families and documenting the trip on his smartphone.
- Copyright: AP
Finnish Prime Minister Juha Sipila says he is ready to open his house for migrants and refugees, and they can move in on 1 January 2016. He told Finnish broadcaster YLE on Saturday that his family has a house in central Finland that they no longer use since moving to Helsinki.
Welcome to our live coverage of the migrant crisis. Follow the latest updates from across the continent as thousands of migrants continue their attempts to reach Europe.