Civilians have begun leaving their besieged homes in several areas after Russia agreed to pause bombardments.
A convoy of buses and cars carrying mostly foreign students were evacuated from the northern city of Sumy, while dozens of people left Irpin near Kyiv.
But Ukraine said Russia was violating another humanitarian corridor from the south-eastern city of Mariupol.
Previous attempts to carry out evacuations have failed, with Russia accused of shelling corridors.
The corridors are expected to stay open until 21:00.
Meanwhile the United Nations said more than two million people had fled Ukraine since Russia launched its invasion nearly two weeks ago, more than half of them going to neighbouring Poland.
Ukrainian cities have continued to come under heavy bombardment from Russian forces.
Russian and Ukrainian officials have held several rounds of talks in an attempt to agree humanitarian corridors.
Ukraine had described Russia’s evacuation plans, many of which involve fleeing residents being sent to Russia, as cynical.
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In Sumy, authorities say children were among 10 people killed in artillery strikes late on Monday.
But evacuations began from the city on Tuesday morning, with buses heading towards the city of Poltava, further south and away from the front lines.
Indian and Chinese students were among those evacuated. The Indian government said all of its 694 students in the city had left with the convoy.
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Human rights activists in the city said the first buses contained almost exclusively foreign students.
“People are afraid to go, because the Russian side has already broken its promise to guarantee the corridors,” they told BBC Russian.
Ukrainian officials said an evacuation route from Mariupol was being shelled in violation of the ceasefire agreement.
The Russian army “launched an attack right at the humanitarian corridor” and “did not let children, women and elderly people leave the city,” the defence ministry said on Facebook.
Meanwhile the situation in the city remains desperate.
One resident, Diana, told the BBC it was terrifying.
“It seemed to me like hell,” she said.
“Because constant shelling above you, and at the same time, no electricity, no heat and no water. The coroner service cannot go and just pick [up] the bodies properly because of the shelling.
“And then there is no network, there is no mobile phone connection, you cannot even call an ambulance if you are injured, or if you are giving birth to a child in a shelter, there is no connection at all.”
Residents have been leaving Irpin, north of Kyiv, many of them crossing a badly damaged bridge blown up to prevent a Russian advance.
“The city is almost ruined, and the district where I’m living, it’s like there are no houses which were not bombed,” a young mother told Reuters.
“As of 09:30 [07:30 GMT], more than 150 people have been evacuated and activities are under way [from Irpin],” said Kyiv Region Governor Oleksiy Kuleba, quoted by the agency.
Russia also says it has opened corridors from the cities of Chernihiv, Kharkiv and Kyiv, but Ukraine has not confirmed this.
Routes for corridors from Kyiv and Kharkiv announced by Russia on Monday were dismissed by Ukraine, as they led only to Belarus and Russia.
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