Jordan Withers, nephew of Glenn Thomas - who was one of the 10 Britons on board flight MH17 - said the bodies of victims had been treated "inhumanely".
Some 298 people died after the airline crashed in eastern Ukraine last week.
PM David Cameron is to use a statement to MPs to press for tougher sanctions against Russia over its response.
Pro-Russia separatists, who retain control of the crash site in east Ukraine, have been blamed for downing the jet - reportedly with a missile - on 17 July.
Russia has been accused of providing the rebels with the anti-aircraft system allegedly used in the incident, allegations the Kremlin denies.
On Sunday, the remains of up to 196 plane victims were loaded on to refrigerated rail wagons in Torez, some 15km (9 miles) away from the crash site. A second train arrived there later to take more bodies on board.
Three Dutch investigators have now examined the bodies being held in Torez, with the team's leader, Peter Van Leit, saying storage was "of good quality".
But Mr Withers told the BBC victims' bodies had been loaded on to trains "like cargo".
His uncle, Glenn Thomas, was a World Health Organisation (WHO) media relations coordinator and former BBC journalist, who was travelling to Australia for an international Aids conference when he died in the crash.
"We just want our uncle back... because that is when we can start the grieving process and we can give him the send-off he deserves," Mr Withers said.
Barry Sweeney, the father of Liam Sweeney, who also died in the crash, told the BBC: "Somebody asked me what would I say to Mr Putin.  I said 'do now - talk later - let our relatives... come home and then sort it out later'.
"I'm not going to be political, I'm not going to blame anybody because they are all as bad as each other in their different ways - the Americans, I'm not even blaming the Russians - I'm just thinking we've got to get it right."
Newcastle United manager Alan Pardew has said the forthcoming season will be dedicated to Liam Sweeney and fellow fan, John Alder, who also died in the crash.
Hugo Hoare, whose brother Andrew died in the crash, told the Telegraph he hoped the treatment of the bodies was "humane", but added: "The first thing I thought was what if they are going to use them as a bargaining chip?"
Mr Hoare, 59, a banker, died along with his Dutch wife and their two children.
The last of the 10 Britons who died were named over the weekend.
They included 44-year-old drilling technician Stephen Anderson and law firm partner John Allen, 44, who died alongside his wife and three sons.
Meanwhile, UK air crash investigators have started work after arriving in the Ukrainian capital Kiev.
The UK Department for Transport said the six-strong team, from the Air Accidents Investigation Branch, was providing "expert technical assistance", although it is not known whether they will gain access to the crash site.
Two Metropolitan Police officers are also in Ukraine as part of the UK's disaster victim identification team.
Mr Cameron told Mr Putin in a "frank" phone call on Sunday that the delay in allowing experts access to the crash site was "completely unacceptable and indefensible".
Monitors from the Organisation for Security and Co-operation in Europe (OSCE) have now visited the accident site, but their access to wreckage has been limited by the separatists. Moscow has been accused of not doing enough to pressure them to permit full international access.
Mr Putin has said in a statement on Russian television it was essential to give international experts security to conduct an investigation.
Mr Cameron, who is due to address the Commons, has backed a new UN resolution to guarantee "unfettered access" to the crash site. The UN Security Council will vote on the proposed resolution later on Monday.
The PM is also due to chair a meeting of the National Security Council, where the issue will be discussed.
And the UK is set to push for increased sanctions against Russia at a meeting of EU foreign ministers on Tuesday.
The BBC's political correspondent Vicki Young said there was a "great deal of diplomatic activity" going on at all levels ahead of the expected vote.
A Downing Street spokesman said the pressure on Mr Putin "would be kept up and stepped up".
Mr Cameron has made it clear he will press for more Russian individuals to be listed for travel bans and asset freezes.
And he is seeking further potential bans on companies and banks that are seen to facilitate the continuing conflict in Ukraine, our correspondent said.
However, BBC political correspondent Gary O'Donoghue said Downing Street was frustrated at the lack of appetite among other EU countries for expanding existing sanctions.
Speaking at a press conference in Whitehall, Deputy Prime Minister Nick Clegg said the time had come "for sanctions to be tightened further".
Chancellor George Osborne said fresh sanctions could harm the UK's economy - but warned that not acting could be "much worse".
Ukraine and the pro-Russian separatists have accused each other of downing the Malaysia Airlines Boeing 777, which had been travelling from Amsterdam to Kuala Lumpur.
However, a spokesman for Mr Cameron said the PM had told the Russian leader that "the evidence suggested that pro-Russian separatists were responsible".
US Secretary of State John Kerry said there was overwhelming evidence of Russian complicity in the downing of the plane.
Relatives of passengers killed in the Malaysia Airlines plane crash have called for the bodies of their loved ones to be returned home.