The first civilians to be evacuated from a giant steel plant in Mariupol were expected to arrive later on Monday in the Ukrainian-held city of Zaporizhzhia after an overnight bus journey across the front-line.
Ukraine says hundreds of civilians have been trapped inside the Azovstal plant along with the city's last Ukrainian defenders.
Dozens were able to leave on Sunday in an evacuation organised by the United Nations, the first to leave since President Vladimir Putin ordered the plant barricaded last week.
In Russia, two explosions took place on Monday in Belgorod, a southern region bordering Ukraine, Governor Vyacheslav Gladkov said.
The cause of the blasts was not clear but the Kremlin has accused Ukraine of making cross-border attacks. Gladkov said there were no casualties.
Ukraine does not generally respond to accusations it is behind attacks in Russia, but has described incidents there as "karma".
An effort to organise the evacuation of civilians from other parts of the city, now held by the Russians, ran into delays. Ukraine says 100,000 people are still living among the ruins in desperate conditions after months of Russian siege.
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"Our house is completely destroyed. We had a two-storey building, it's not there anymore. It burned to the ground," said Natalya Tsyntomirska, a Mariupol native who reached Zaporizhzhia on Monday in a funeral service van.
After being forced to abandon an assault on Kyiv at the end of March, Russia has launched a major new offensive in eastern Ukraine. For its part, Kyiv hopes a massive influx of Western military aid will allow it to repel that assault and then turn the tide with a counter-attack.
The Russian offensive is focused on the eastern Donetsk and Luhansk provinces, parts of which were already held by Russian-backed separatists before the invasion. Russian troops are now trying to encircle a large Ukrainian force there, attacking from three directions with massive bombardment along the front.
Ukraine's military said on Monday Russian forces were trying to take over the frontline Luhansk province town of Rubizhne, and prepare an assault on nearby Sievierodonetsk.
The heaviest clashes were taking place around Popasna, further south. Shelling was so intense it was not possible to collect the bodies, said regional Governor Serhiy Gaidai.
"I don't even want to speak about what's happening with the people living in Popasna, Rubizhne and Novotoshkivske right now. These cities simply don't exist anymore. They have completely destroyed them."