Tory leadership rivals Boris Johnson and Jeremy Hunt have unveiled pledges on immigration and education.
Frontrunner Mr Johnson has promised to deliver an Australian-style points-based immigration system if he becomes prime minister.
And Foreign Secretary Mr Hunt said he would cancel the tuition fee debts of young entrepreneurs who start businesses and employ people.
The winner of the contest will take over from Theresa May on 24 July.
In the Conservative Party digital hustings, broadcast on the party's Facebook page and on Twitter on Wednesday evening, Mr Johnson said he was "open to talent, open to immigration" but he said it "should be controlled".
He called for an Australian-style-points-based system, considering factors including whether an immigrant has a firm job offer before arrival and their ability to speak English.
Mr Johnson said he would also block the ability to claim benefits immediately when someone arrives in the UK.
"We must be much more open to high-skilled immigration such as scientists, but we must also assure the public that, as we leave the EU, we have control over the number of unskilled immigrants coming into the country," he said.
This would restore public faith and democratic control in the immigration system, while being "tougher on those who abuse our hospitality", he added.
The Immigration and Social Security Co-ordination (EU Withdrawal) Bill is currently at report stage and contains plans to end the existing automatic preference for EU citizens.
However, the bill does not set out what the new immigration system will look like from 2021, and a consultation to shape this is not due to report back to Parliament until the end of this year.
Madeleine Sumption from the Migration Observatory says Australia has used its points system in order to "create a very liberal system to push up the number of migrants".
She explained a point system similar to Australia's "would have the effect of being a little bit more open to high-skilled migration from outside the EU and more restrictive to low-skilled migration from inside the EU."
'Turbo-charge our economy'
Under Mr Hunt's new proposal, anyone who creates a new business which employs more than 10 people for five years would have their university tuition fee debts written off.
Mr Hunt, who has stressed his background as an entrepreneur, said he wanted to give young people the confidence to go into business for themselves.
"If we are to turbo-charge our economy and take advantage of Brexit, we need to back the young entrepreneurs who take risks and create jobs," he said.
"I started my own business, I still use the lessons that experience taught me - focus, drive and the art of negotiation - every single day.
"I want more young people to have the confidence to take the decision to start their own business, so we create wealth and start thriving as a country again."
The current Prime Minister Theresa May has refused to promise unconditional support for her successor's Brexit plan.
When asked if she would back whatever Brexit outcome her successor was able to achieve - including a no-deal Brexit - she said that amounted to inviting her to say she would agree with "whatever happens in future".
The ringmaster or the safe-ish pair of hands?
Smart? Intelligent? Surely, yes we would all want a clever politician in charge, someone who grasps complexity and is willing to put in the hours to solve the country's problems.
But without resorting to the insult that the inhabitants of Westminster would have been on stage and screen if only they'd been more aesthetically blessed, politics also has an element of performance about it.
It is a very serious form of showbiz if you like. Its most senior figures become famous, even if often for all the wrong reasons.
Political parties want their leaders to have "cut through", to use the terrible jargon, not just to communicate clearly, but to make people want to listen.
The Institute for Fiscal Studies has estimated that Mr Hunt's proposals, including a corporation tax cut and an increase in defence spending, would cost a total of between £37-£68bn.
It concluded the foreign secretary's plans would leave no scope to relieve the pressure on other spending departments without tax rises or risking higher borrowing.
The IFS also analysed Mr Johnson's tax plan, which includes increasing the £50,000 threshold for the higher rate of income tax, and concluded it would cost "many billions" and benefit the wealthy most.
Meanwhile, Mr Hunt spoke for the first time about the death of his sister in a "terrible accident" when he was only two years old.
"I was too young to ever remember it but I do know it affected my parents," he told ITV's Peston.
"I wouldn't claim this is something that had a big emotional effect on me personally."
Each candidate is vying for the votes of the 160,000 or so Conservative Party members who will vote for the next party leader and therefore prime minister.
The rivals used the digital hustings to set out some of their plans:
Mr Hunt said if the UK gets to October without the prospect of a deal, "we will leave without a deal".
Mr Johnson also repeated his pledge to get the UK out of the EU on 31 October, but he thinks the chances of a no-deal Brexit happening are a "million to one".
However, in an interview with Conservative Home, Mr Johnson said every member of his cabinet would have to be "reconciled" with the policy of leaving on 31 October - with or without a deal.
The Work and Pensions Secretary Amber Rudd has said it is too "premature" to say whether she would serve in a cabinet under Mr Johnson. Ms Rudd is supporting Jeremy Hunt to be party leader.
On an election
Mr Hunt said it was essential the Conservatives had delivered Brexit before a general election, otherwise the party "will be thrashed".
Asked about proroguing Parliament, Mr Johnson said: "It would be absolutely crazy for any of us to think of going to the country and calling a general election before we get Brexit done."
When pushed, he added: "I'm not attracted to archaic devices like proroguing."
On trade rules
Mr Hunt dismissed rival Mr Johnson's claims that a mechanism known as GATT 24 could be used to prevent tariffs if there was a no-deal Brexit.
He said: "I think we've got to knock this GATT 24 thing on the head. You can only get an agreement not to introduce tariffs if both sides agree to that."
On EU citizens' rights
Mr Hunt said he would give full citizens' rights to the three million EU nationals living in the UK, even if the UK left without a deal.
Mr Johnson said he was "absolutely in favour" of protecting the rights of EU citizens in the UK.
The foreign secretary said he would back both the third runway at Heathrow and the HS2 high-speed rail link.
Mr Johnson said it had to be recognised that there was "huge pressure" on the South East, so more infrastructure was needed.
Mr Johnson said the NHS would be "free to everybody at the point of use" under his premiership.
He ruled out a pay-for-access NHS, even as a result of a post-Brexit trade deal with the US.
Westminster's public spending watchdog, the public accounts committee, said NHS recruitment and school funding must be the first priorities of the new prime minister, warning that Brexit had been overshadowing other issues.
Committee chairwoman Meg Hillier said education funding was a "crisis" and described NHS staffing shortages as "unsustainable".
The candidates are set to face each other at an ITV debate on 9 July and at an event hosted by the Sun newspaper and talkRADIO on 15 July.