As Russia prepares to annex a swathe of Ukrainian territory the size of Portugal after staging what it calls referendums in four occupied provinces, Ukrainians who have been able to escape describe an exercise that would be funny if it were not so menacing.

"They can announce anything they want. Nobody voted in the referendum except a few people who switched sides. They went from house to house, but nobody came out," Boyko said.

He, his wife and their two children had arrived at the aid centre in the parking lot of a home improvement store in Ukrainian-held Zaporizhzhia city the previous day, after waiting for two days before Russian forces abruptly allowed them out through the last checkpoint.

Those fleeing Russian-held territory say the so-called referendum has been carried out by men with guns forcing people to cast ballots in the street. The biggest fear is that, as soon as Moscow declares the territory to be Russia, it will immediately start press-ganging men to fight in its forces.

For now, Russian forces have been letting some people out of occupied parts of Kherson and Zaporizhzhia provinces through the one checkpoint. No one knows how long the route will stay open, especially for draft-aged men.

Hundreds of people arrived on Tuesday in cars and mini-vans after being suddenly allowed out. On Wednesday morning some were still at the centre, with nowhere to go, trying to arrange accommodation after spending the night in a school.

An air raid siren sounded, and a downpour drenched the parking lot of the Epicenter home improvement store that hosts the reception center in plastic shelters. A U.S.-based charity, World Central Kitchen, was providing hot meals inside a tent.

"A lot of people are just leaving everything behind. There are places that are completely deserted," said Boyko. "Everybody wants to be in Ukraine, and this is why everybody is leaving. Over there is a lawless place. Entire villages are leaving."

He said officers of the FSB, Russia's internal security service, had told him and others at the last checkpoint that if they leave for Ukraine they would never be permitted to return. He did not know if draft-age men still were being allowed out.

"The line of vehicles was so long you could not see the end of it," recounted another man, Andriy, 37, who declined to give his last name, standing by the yellow, mud-spattered minibus in which he arrived with his wife, two children and parents.

"Seventy percent of people are leaving because of the referendum. There was no light, no gas, and no work and all of a sudden, you get the referendum," said the agricultural worker from Beryslav, in Kherson province. "It's complete nonsense. I don't know a single person among those I know who voted."

He said he saw passers-by forced to fill out ballots on their knees at a Bereslav crossroad.

Russia says voting has been voluntary and turn-out has been high. Pro-Russian officials have published what they describe as results showing overwhelming support for annexation. Kyiv and Western countries call the exercise a complete sham, aimed at justifying the annexation of territory seized by force.

"If I came to your home and told you, 'Now this place is mine,' what would you do?" chimed in Andriy's 60-year-old father Viktor.

"Would you hand it over? No, you would chase them off with a pitchfork. The Russians are morally ugly. This is all awash in blood."

(Reporting by Jonathan Landay,; Editing by Peter Graff and Angus MacSwan)

By Jonathan Landay