Residents have been told to leave Ukraine's eastern city of Slovyansk, a day after two people were killed in a Russian attack on a market.
Civilians boarded minibuses on Wednesday morning as Russian forces advanced from the north and east.
UK defence intelligence said there was a "realistic possibility" the battle for Slovyansk would be the next key contest in the struggle for the Donbas.
Russian forces were likely to be 16km (10 miles) north of the city, it said.
Donetsk governor Pavlo Kyrylenko said the Russians had turned the entire Donetsk region into a hotspot where it was dangerous for a civilian to remain. "I call on everyone to evacuate, evacuation saves lives," he said on his social media feed.
Russia switched its focus to Ukraine's eastern Donbas region in March, after a failed bid to capture the capital Kyiv. It has taken control of Luhansk region, to the north and east of Slovyansk, and has now pushed on to the rest of Donetsk. The industrial Donbas embraces Donetsk and Luhansk.
Russian commentators said the fall of the last city in Luhansk had cleared the way for an advance towards towns to the west and south.
However, Ukraine's regional chief in Luhansk, Serhiy Haidai, said the military continued to frustrate Russian forces on the border with Donetsk region, particularly around the main road from Lysychansk to Bakhmut.
Russian military expert Vladislav Shurygin told Izvestia newspaper that it would take the army several days to clear mines, but the next large-scale battles would be for Slovyansk and neighbouring Kramatorsk, as well as pushing Ukrainian forces out of range of the city of Donetsk.
Slovyansk was seized by Russian-backed forces when the war first erupted in 2014, but recaptured two and a half months later. As residents remembered the anniversary of the army's success on Tuesday, Russian artillery pounded the city, setting the market on fire.
Elsewhere in Ukraine, a civilian was killed early on Wednesday in a Russian cruise missile strike on a cafe in the southern Mykolaiv region, local leaders said. And in Kharkiv in the north a security guard was killed when a university building was shelled overnight.
Irish leader Micheál Martin visited some of the towns around Kyiv on Wednesday which had suffered serious damage in Russia's bombardment. The Taoiseach also went to the site of a mass grave in Bucha where residents had been murdered.
Although Russian forces pulled back from around Kyiv a month into the invasion, Russian Security Council chief Nikolai Patrushev has reiterated that the Kremlin's main aim is still to replace Ukraine's leaders and capture a broad swathe of its territory.
A close ally of the Russian leader, he repeated President Vladimir Putin's earlier claims of "de-Nazifying" and demilitarising Ukraine, saying that those goals would be achieved despite Western military aid to Kyiv.
In reality Ukraine's President Volodymyr Zelensky has no connection to the far right: he is a former comedian, democratically elected and ethnically Jewish.
A measure of Mr Putin's thinking at the time of the invasion was revealed in a French documentary last week, which broadcast a tense nine-minute phone call in which he told President Emmanuel Macron that President Zelensky was not democratically elected and came to power during a bloodbath when people were burned alive.
Russian Foreign Minister Sergei Lavrov said on Wednesday that broadcasting the call was a breach of diplomatic etiquette, although Russia had nothing to be ashamed of.