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The men, identified as pro-Ukrainian fighters Shaun Pinner and Aiden Aslin, addressed the camera to ask British authorities to arrange a swap for Kremlin ally Viktor Medvedchuk, who is detained by Ukraine.
It was not clear from the footage how freely the two men were speaking, and the broadcast did not say where or by whom they were being held.
Mr Pinner’s family issued a statement saying the former British Army soldier was not a volunteer or mercenary but was serving officially with the Ukrainian marines.
“Shaun enjoyed the Ukrainian way of life and considered Ukraine as his adopted country,” it said. The family said Mr Pinner moved to Ukraine in 2018 and met his wife there.
The family said they were working with the Foreign Office and the relatives of Mr Aslin to ensure that their rights under the Geneva Conventions were respected.
Mr Aslin’s mother told the Daily Telegraph that she recognised her son in an earlier appearance on Russian television and called on Moscow’s forces to treat him humanely.
In footage broadcast on the Rossiya 24 state TV channel on Monday, Shaun Pinner addresses the UK Prime Minister and appears to ask for himself and a fellow British prisoner of war, Aiden Aslin, to be swapped for Viktor Medvedchuk, who has been held in Ukraine.
“Hi Mr Boris Johnson,” says the 48-year-old former Royal Anglian soldier, who appears tired in the video.
“I understand that Mr Medvedchuk has been detained and we look to exchange myself and Aiden Aslin for Mr Medvedchuk.
“Obviously, I’d really appreciate your help in this matter and pushing this agenda.”
He also says he has been “treated well” and “fed, watered”.
In a separate clip, Mr Aslin, 28, is heard saying: “If Boris Johnson really does care like he says he does about British citizens, then he would help pressure [Ukrainian President Volodymyr] Zelenskyy to do the right thing and return Viktor to his family and return us to our families.”
It is unclear if the men were speaking under duress.
In the latest footage on Russia 24 television, a state TV journalist showed the two prisoners a video published by Mr Medvedchuk’s wife, Oksana Marchenko, in which she demanded her husband be released.
A video separately published by Ukraine showed Mr Medvedchuk asking the Kremlin to swap him for Ukrainians and soldiers defending Mariupol.
Mr Zelenskyy had earlier offered to exchange Mr Medvedchuk, 67, for Ukrainians being held in Russia.
Mr Medvedchuk is under sanctions over what western governments described as his close ties to the Kremlin. Britain said he had been identified as a possible pro-Russian puppet leader in Ukraine.
Asked about a potential exchange last week, Kremlin spokesman Dmitry Peskov said Mr Medvedchuk was not a Russian citizen and said he did not know if the businessman wanted Moscow to intercede.
In a second video, Mr Pinner appeared to be questioned by a Russian journalist about how he was captured.
He described having been in a factory area in Mariupol, the besieged southern port that has seen some of the war’s heaviest fighting, and where Russia claims hundreds of foreign fighters are among the city’s defenders.