Ukrainian officials say people in the city of Mariupul have been taken against their will to Russia.
The news of the 'abduction' comes as Russian forces attacked an art school in the city which was being used as a shelter from constant shelling and left it in ruins
People are believed to be trapped under the rubble where it is thought 400 people had taken refuge but there was no details on the number of casualties.
The city's council said thousands had been 'forcibly deported' to Russia as they fight to protect their homeland.
The Mariupol authority claimed Russia "illegally took people" from Livoberezhniy district and from a sports club property which was being used as a bomb shelter.
The Kremlin has claimed buses carrying "refugees" from Mariupol began to arrive to Russia on Tuesday.
Mariupol's mayor Vadym Boychenko compared the situation to Nazis removing civilians in the Second World War.
He said: "It is known that the captured Mariupol residents were taken to filtration camps, where the occupiers checked people's phones and documents.
"After the inspection, some Mariupol residents were redirected to remote cities in Russia; the fate of others remains unknown.
"What the occupiers are doing today is familiar to the older generation, who saw the horrific events of World War Two, when the Nazis forcibly captured people."
The mayor said it was difficult to imagine that deportation could happen in the 21st Century in addition to destroying the city
Ukrainian President Volodymyr Zelensky, meanwhile, said Russia's siege of the port city of Mariupol was "a terror that will be remembered for centuries to come".
"Over the past week, several thousand Mariupol residents were deported onto the Russian territory," the city council said in a statement on its Telegram channel late on Saturday.
Russian news organisations have said buses have carried several hundred people Moscow calls refugees from Mariupol to Russia in recent days.
Many of Mariupol's 400,000 residents have been trapped for more than two weeks as Russia seeks to take control of the city, which would help secure a land corridor to the Crimea peninsula.
President Vladimir Putin calls the assault on Ukraine a "special operation" aimed at demilitarising the country and rooting out people he terms dangerous nationalists.
Western nations call it an aggressive war of choice and have imposed punishing sanctions on Russia aimed at crippling its economy.
The Mariupol bombardment has left buildings in rubble and severed central supplies of electricity, heating and water, according to local authorities.
Rescue workers were still searching for survivors in a Mariupol theatre that local authorities say was flattened by Russian air strikes on Wednesday. Russia denies hitting the theatre.
Air raid sirens sounded across Ukrainian cities on Sunday and Russia's defence ministry said cruise missiles were launched from ships in the Black Sea and Caspian Sea, as well as hypersonic missiles from Crimean airspace.
Hypersonic missiles were deployed by Russia for the first time in Ukraine on Saturday, in a strike which Moscow claimed destroyed a large underground depot for missiles and aircraft ammunition.
A spokesperson for the Ukrainian Air Force Command confirmed the attack in the western Ivano-Frankivsk region, but said the Ukrainian side had no information on the type of missiles used.
Ukraine's Deputy Prime Minister Iryna Vereshchuk said seven humanitarian corridors would open on Sunday to enable civilians to leave frontline areas.
