The Russian army has captured several towns and villages in recent days, even as Moscow scrambles to fight off a Ukrainian counterattack into its western Kursk region.
Civilians in Myrnograd -- less than 10 kilometres (six miles) from the front line -- told AFP that increased shelling had finally prompted some to leave, two and a half years after Russia launched its full-scale invasion.
"I need to leave, because the situation is really getting worse. Every day -- not even every day, but every hour -- is like an avalanche," said Maksym, a 40-year-old mine worker.
A recent strike hit the nine-storey block of flats where he lives, blowing out the windows.
"Thank God I wasn't home but I decided to leave, because life is precious," he said.
AFP reporters saw civilians watching as houses burned after a recent Russian shelling attack on the small town.
The remains of a half-destroyed residential building stood surrounded by rubble and a burnt-out car was crumbling by the side of the road.
Firefighters were tackling a blaze at another house hit in a recent barrage, as its owner stood outside, shirtless.
Russian troops are fighting to reach the key logistics hub of Pokrovsk, a strategically important city five kilometres (three miles) west of Myrnograd.
Officials on Monday ordered families with children to evacuate Pokrovsk and the surrounding areas, where they said more than 50,000 people still live.
Asked whether he would return one day, Maksym said: "I'd like to believe so."
Galyna, 74, on Wednesday was helping her granddaughter leave for the comparative safety of central Ukraine.
"My granddaughter is scared. I feel sorry for the children," she said.
Regional officials said they had set-up a temporary accommodation centre for families forced to abandon their homes but with nowhere else to go.
Russia on Wednesday claimed its latest territorial advance, with the defence ministry saying its forces had captured the town of Zhelanne, around 20 kilometres (12 miles) to the southeast.
Moscow claims to have annexed the industrial Donetsk region, as well as three others in eastern and southern Ukraine, despite not having full control over any of them.
The Donetsk region has been at the centre of the war between Russia and Ukraine since 2014, when Moscow-backed separatists tried to seize control of the Donbas region and secede from Kyiv.
Separately, Ukraine's parliament voted on Wednesday to join the International Criminal Court (ICC), as Kyiv seeks to bring Russia to justice over war crimes it is alleged to have committed throughout its invasion.
Ukraine signed the Rome Statute that founded the court in 2000, but had not ratified it on fears Ukrainian soldiers could face prosecution.
Ukrainian military units disputed Russia's claim that it had taken control of the town of New York, one of its key targets in recent months.
Moscow's defence ministry said Tuesday its forces had seized it in a recent advance.
Kyiv might have hoped its shock border incursion into Kursk, now in its third week, would force Moscow to divert troops from other parts of the frontline.
It claims to have captured dozens of settlements and more than 1,000 square kilometres of territory in the unprecedented cross-border assault.
So far there has been little sign that fighting on the frontlines in Ukraine's east has subsided, however.
Both countries also launched attempted overnight drone strikes aimed at Kyiv and Moscow.
Russia said it destroyed 45 drones -- 11 of them headed for the Russian capital.
"This is one of the largest ever attempts to attack Moscow with drones," Moscow Mayor Sergei Sobyanin said.
Ukraine's air force said it detected 72 air targets over Ukraine and downed 50 drones and one missile,