Dozens of civilians have been evacuated from Mariupol to both Russia-controlled and Ukraine-controlled territory after weeks under siege.
Some have left the Azovstal steelworks, the last hold-out of Ukrainian troops in the strategically significant city.
Russia said dozens of civilians have arrived in a village it controls.
Ukrainian President Volodymyr Zelensky said a large group is also on its way to Zaporizhzhia, which Ukraine maintains control of.
"The first group of about 100 people is already heading to the controlled area," he tweeted. "Tomorrow we'll meet them in Zaporizhzhia. Grateful to our team! Now they, together with UN, are working on the evacuation of other civilians from the plant."
The United Nations confirmed it was involved with the operation, alongside the Red Cross and the "parties to the conflict".
The evacuation convoy arrived on Saturday morning, the UN said - but did not give details on where people were being taken or how many had left, saying that sharing details could jeopardise the safety of the operation.
The first civilians managed to leave the Azovstal plant on Saturday, but their fate was unclear for most of the following day.
Now, the Russian defence ministry says some 80 civilians have left the besieged steelworks and were taken to Bezimenne, a village in Russian-held territory in Ukraine. They are being provided with medical care and supplies, it said.
In a statement carried by Russian media, the ministry also said that civilians who wanted to leave for Ukraine-controlled areas "have been handed over to representatives of the UN and the International Committee of the Red Cross".
Reuters news agency reported that one of its photographers stationed in the village had seen more than 50 arrive in a convoy, and reported that UN staff were in the area.
Mariupol has come under intense bombardment from Russian artillery for weeks.
The Azovstal steelworks - where Ukraine's last defenders are holed up with some remaining civilians - is a vast industrial area with a network of underground tunnels, making its capture difficult for Russian forces.
Last month, after declaring that the southern Ukrainian city had been captured, Russian President Vladimir Putin ordered his troops to seal off the steelworks "so that a fly cannot not pass through."
The devastation across Mariupol has been some of the most severe in the conflict, with much of the city reduced to rubble and a widespread humanitarian crisis. Fleeing residents have spoken about a lack of food and water, and bodies of the dead being left in the streets.
Speaking to the BBC on Saturday, Mariupol mayor Vadym Boychenko said: "The citizens who left the city say that hell exists and it's in Mariupol."
With most of the city now under Russian control, the focus remains on the conflict-hit steelworks.
Shortly after news of Sunday's wider evacuation project emerged, Mariupol officials said that evacuation from areas of the city other than the Avozstal industrial plant had been put on hold until Monday morning.
- BBC