(Adds Zelenskiy quote; details from Ukrainian military, deputy defence minister)
* Ukrainian forces evacuate from Mariupol plant
* Ukraine regiment says it is saving lives of troops inside
* Evacuees arrive in Russian-controlled town
* Putin responds calmly to Finland, Sweden NATO move
By Natalia Zinets
KYIV/MARIUPOL, Ukraine, May 16 (Reuters) - Troops holed up in the last Ukrainian stronghold in the besieged port of Mariupol began evacuating on Monday, appearing to cede control of the once prosperous city to Russia after months of bombardment. Ukraine's deputy defence minister said 53 injured troops from the Azovstal steelworks were taken to a hospital in the Russian-controlled town of Novoazovsk, some 32 kilometres (20 miles) to the east.
Another 211 people were taken to the town of Olenivka, in an area controlled by Russian-backed separatists, Deputy Defence Minister Anna Malyar said. All of the evacuees will be subject to a potential prisoner exchange with Russia, she added.
Reuters saw five buses carrying troops from Azovstal arrive in Novoazovsk late on Monday. Some of the evacuated troops were wounded and carried out of the buses on stretchers. Some 600 troops were believed to have been inside the steel plant.
"We hope that we will be able to save the lives of our guys," Ukraine President Volodymyr Zelenskiy said in a late night address. "There are severely wounded ones among them. They're receiving care. Ukraine needs Ukrainian heroes alive."
Ukraine's military said it had "ordered the commanders of the units stationed at Azovstal to save the lives of the personnel" and that troops there had fulfilled their combat mission.
Efforts to rescue troops still inside were under way, the military added. It did not say how many troops remained.
Ukrainian troops say they held out in Azovstal for 82 days, buying time for the rest of Ukraine to battle Russian forces and secure Western arms needed to withstand Russia's assault.
But the evacuation likely marked the end of the longest and bloodiest battle of the Ukraine war and a significant defeat for Ukraine. Mariupol is now in ruins after a Russian siege that Ukraine says killed tens of thousands of people in the city.
Since Russia launched its invasion in February, Mariupol's devastation has become a symbol both of Ukraine's ability to withstand Russia's invasion and of Russia's willingness to devastate Ukrainian cities that hold out.
The evacuation came hours after Russia said it had agreed to evacuate wounded Ukrainian soldiers to a medical facility in Novoazovsk.
Azovstal's last defenders had been holding out for weeks in bunkers and tunnels built deep underground to withstand nuclear war. Civilians were evacuated from inside the plant, one of the largest metallurgical facilities in Europe, earlier this month.
The wife of an Azov Regiment member described conditions at the plant earlier on Monday: "They are in hell. They receive new wounds every day. They are without legs or arms, exhausted, without medicines," Natalia Zaritskaya said.
PUTIN CLIMBDOWN OVER NATO
Russian President Vladimir Putin appeared earlier on Monday to climb down from threats to retaliate against Sweden and Finland for announcing plans to join the U.S.-led NATO military alliance.
"As far as expansion goes, including new members Finland and Sweden, Russia has no problems with these states - none. And so in this sense there is no immediate threat to Russia from an expansion to include these countries," Putin said.
The comments appeared to mark a major shift in rhetoric, after years of casting NATO enlargement as a direct threat to Russia's security, including citing it as a justification for the invasion of Ukraine itself.
Just hours before Putin spoke, Russia's deputy foreign minister, Sergei Ryabkov, said Finland and Sweden were making a mistake that would have far-reaching consequences: "They should have no illusions that we will simply put up with it."
Putin said NATO enlargement was being used by the United States in an "aggressive" way to aggravate an already difficult global security situation, and that Russia would respond if the alliance moves weapons or troops forward.
"The expansion of military infrastructure into this territory would certainly provoke our response. What that (response) will be - we will see what threats are created for us," Putin said.
Finland and Sweden, both non-aligned throughout the Cold War, say they now want the protection offered by NATO's treaty, under which an attack on any member is an attack on all.
"We are leaving one era behind us and entering a new one," Swedish Prime Minister Magdalena Andersson said, announcing plans to formally abandon militarily non-aligned status - a cornerstone of national identity for more than 200 years.
Kjell Engelbrekt, professor of political science at the Swedish Defence University, said Moscow now had few military options left to follow through on its previous "very assertive" rhetoric demanding the Nordics never join NATO.
A senior U.S. official, speaking on condition of anonymity, said Washington had seen no indications Russia was moving troops or equipment closer to the border with Finland.
UKRAINE TROOPS REACH BORDER
Moscow calls its invasion a "special military operation" to rid Ukraine of fascists, an assertion Kyiv and its Western allies say is a baseless pretext for an unprovoked war.
Nearly three months old, Russia's invading forces have run into apparent setbacks, with troops forced out of the north and the environs of Kyiv in late March. A Ukrainian counterattack in recent days has driven Russian forces out of the area near Kharkiv, the biggest city in the east.
Ukraine's defence ministry said on Monday troops had advanced all the way to the Russian border, about 40 km north of Kharkiv.
The successes near Kharkiv could let Ukraine attack supply lines for Russia's own main offensive, grinding on further south in the Donbas region, where Moscow has been launching mass assaults for a month yet achieving only small gains.
In a video message, Zelenskiy hailed the achievement and thanked the troops: "I am very grateful to you from all Ukrainians, from everyone, from myself, from my family, my gratitude is unlimited."
(Reporting by Natalia Zinets in Kyiv and a Reuters journalist in Mariupol; Additional reporting by Ron Popeski and Reuters bureaux; Writing by Peter Graff; Editing by Catherine Evans and Cynthia Osterman)