Efforts were underway on Tuesday to rescue the last of the defenders inside the Azovstal steel plant in the ruined city of Mariupol after Ukrainian officials said the fighters had "completed their mission" and there was no way to free the plant by military means.

The Ukrainian military avoided using the term "surrender" to describe the effort to pull out of the steel plant to save as many lives as possible. Officials planned to keep trying to save an unknown number of fighters who stayed behind. It was unclear if soldiers evacuated to Russian-controlled areas would be considered prisoners of war.

Ukraine's deputy defence minister said more than 260 fighters, including some badly wounded, were evacuated from the plant on Monday and taken to areas under Russia's control.

"The work to bring the guys home continues, and it requires delicacy and time," said Ukrainian president Volodymyr Zelensky.

He said the evacuation to separatist-controlled territory was done to save the fighters who endured weeks of Russian assaults in the maze of underground passages below the plant.

"Ukraine needs Ukrainian heroes to be alive. It's our principle," he said.

Russian forces were pounding targets in the industrial heartland of eastern Ukraine known as the Donbas, and the death toll kept climbing with the war set to enter its 12th week.

Ukraine made a symbolic gain when its forces reportedly pushed Russian troops back to the Russian border in the Kharkiv region.

Video showed Ukrainian soldiers carrying a post that resembled a Ukrainian blue-and-yellow-striped border marker and then posing next to it.

The Ukrainian border service said the video showing the soldiers was from the border "in the Kharkiv region", but would not elaborate, citing security reasons. It was not immediately possible to verify the exact location.

Zelenskyy thanked the soldiers, saying in a video message: "I'm very grateful to all the fighters like you."

Ukraine deputy defence Minister Hanna Maliar said 53 seriously wounded fighters were taken from the Azovstal plant to a hospital in Novoazovsk, east of Mariupol.

Another 211 fighters were evacuated to Olenivka through a humanitarian corridor. She said an exchange would be worked out for their return home.

Before the evacuations began, the Russian defence ministry announced an agreement for the wounded to leave the mill for treatment in a town held by pro-Moscow separatists.

After nightfall on Monday, several buses pulled away from the steel mill accompanied by Russian military vehicles. Maliar confirmed the evacuation had taken place.

She said the "defenders of Mariupol" had fulfilled all their tasks, and it was impossible to "unblock Azovstal by military means".

"Mariupol's defenders have fully accomplished all missions assigned by the command," Maliar said.

The commander of the Azov Regiment, which led the defence of the plant, said in a pre-recorded video message released on Monday that the regiment's mission had ended with as many lives saved as possible.

"Absolutely safe plans and operations don't exist during war," Lt Col Denis Prokopenko said.

Elsewhere in the Donbas, the eastern city of Sievierdonetsk came under heavy shelling that killed at least 10 people, said Serhiy Haidai, the governor of the Luhansk region.

In the Donetsk region, governor Pavlo Kyrylenko said on Facebook that nine civilians were killed in shelling.

The western Ukrainian city of Lviv was rocked by loud explosions early on Tuesday. Witnesses counted at least eight blasts accompanied by distant booms. An Associated Press team in Lviv, which was under an overnight curfew, said the sky west of the city was lit up by an orange glow.

The chairman of the Lviv regional military administration said the Russians fired on military infrastructure in the Yavoriv district. The city of Yavoriv is less than 10 miles from the Polish border.

Ukrainian border guards said they also stopped a Russian attempt to send sabotage and reconnaissance troops into the Sumy region, some 90 miles north-west of Kharkiv.

In another setback for Moscow, Sweden's decided to seek NATO membership following a similar decision by neighbouring Finland. That is a historic shift for the countries, which have been nonaligned for generations.

Swedish prime pinister Magdalena Andersson said her country would be in a "vulnerable position" during the application period.

She urged her fellow citizens to brace themselves for disinformation or other attempts to divide them.

President Recep Tayyip Erdogan of Turkey, a NATO member, has objected to allowing Sweden and Finland to join NATO, saying they failed to take a "clear" stance against Kurdish militants and other groups that Ankara considers terrorists, and imposed military sanctions on Turkey.

All 30 current NATO members must agree to let the Nordic neighbours join.