A residential building, which was damaged during the Ukraine-Russia conflict, in Mariupol. Reuters

Russian forces pushed deeper into Ukraine's besieged and battered port city of Mariupol on Saturday, where heavy fighting shut down a major steel plant and local authorities pleaded for more Western help.

The fall of Mariupol, the scene of some of the war's worst suffering, would mark a major battlefield advance for the Russians, who are largely bogged down outside major cities more than three weeks into the biggest land invasion in Europe since World War II.

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"Children, elderly people are dying. The city is destroyed and it is wiped off the face of the earth,” Mariupol police officer Michail Vershnin said from a rubble-strewn street in a video addressed to Western leaders that was authenticated by The Associated Press.

Children walk near residential buildings, which were damaged during the Ukraine-Russia conflict, in Mariupol. Reuters

Mariupol has been the site of some of the war’s greatest suffering. Ukrainian President Volodymyr Zelensky said early on Sunday that what Russian forces did to the peaceful city of 430,000 "is a terror that will be remembered for centuries to come.”

In his nightly address to the nation, Zelensky said Ukraine is interested in peace, and that ongoing talks with Russia are "not simple or pleasant, but they are necessary.” He has requested to meet directly with Russian President Vladimir Putin, but a meeting like that hasn't happened.

Zelensky said Russia's siege of Mariupol was "a terror that will be remembered for centuries to come", while authorities in the strategic port city said thousands of residents there had been taken by force to Russia.

Service members of pro-Russian troops drive an armoured vehicle in Mariupol. Reuters

Zelensky said the siege of Mariupol will go down in history for what he’s calling war crimes by Russia’s military.

"To do this to a peaceful city, what the occupiers did, is a terror that will be remembered for centuries to come,” he said early Sunday in his nighttime video address to the nation.

Details also began to emerge Saturday about a rocket attack that killed as many as 40 marines in the southern city of Mykolaiv the previous day, according to a Ukrainian military official who spoke to The New York Times.

Russian forces have already cut Mariupol off from the Sea of Azov, and its fall would link Crimea, which Russia annexed in 2014, to eastern territories controlled by Moscow-backed separatists. It would mark a rare advance in the face of fierce Ukrainian resistance that has dashed Russia's hopes for a quick victory and galvanized the West.

A local resident holds a dog outside a residential building in Mariupol. Reuters

Ukrainian and Russian forces battled over the Azovstal steel plant in Mariupol, Vadym Denysenko, adviser to Ukraine’s interior minister, said. "One of the largest metallurgical plants in Europe is actually being destroyed,” Denysenko said in televised remarks.

The Mariupol city council claimed hours later that Russian soldiers had forcibly relocated several thousand city residents, mostly women and children, to Russia. It didn't say where, and AP could not immediately confirm the claim.