Those attempting to escape includes survivors from Mariupol, and others who were captured and taken to Russian regions.
"There is an impressive grassroots organisation on many levels. There are groups that collect clothes and toys for children," an activist told iNEws.
"Others put people up in their homes for a few nights, buy train tickets and take them to the station. We are sharing messages and passing people on to groups in other cities, who are helping them get over the border."
It comes as Vladimir Putin broadcasted a Victory Day speech on Russian TV, which was interrupted by hackers.
The messages on-screen told viewers and Putin's supporters that their hands are 'covered in blood' for the the invasion of Ukraine
Putin used the speech to accuse Western countries of leaving him with no choice but to accelerate the invasion.
He said the military action was a necessary response to Western policies and alliances, describing it as a way to ward off aggression.
The White House say the speech showed Putin is lacking direction, after he targeted NATO over various conspiracies inflicting Russia.
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