Many waited for hours in freezing conditions to leave Ukraine after Russian President Vladimir Putin ordered an invasion.
Iryna, 36, and her mother set out from Kyiv on Thursday with her two girls aged 2 and 4, before crossing into Ubla in Slovakia.
"We left my husband there, so he is still there supporting our government," she said on Friday at a hotel in the border town of Snina.
"We pray for Ukraine and I hope everything will be fine," she said.
In Poland, which has the region's largest Ukrainian community of about one million people, authorities said wait times to cross the border ranged from six-12 hours in some places.
At Medyka in the south of Poland, 85km from Lviv in western Ukraine, roads were packed with cars, police directing traffic and people hugging loved ones after they arrived on the Polish side.
An internet map site showed a third of the way congested with heavy traffic.
Ukrainian rules restrict men aged 18-60, who could be conscripted, from crossing the border.
Marta Buach, 30, from Lviv, said her husband was not allowed to cross with her.
"In Lviv it is okay but in other cities it is really a catastrophe. Kyiv was shelled, other small cities were shelled, we were hearing bombing everywhere.
"I think it is only a matter of time for it to be as dangerous as other cities," she said.
United Nations aid agencies say the war could drive up to five million people to flee abroad.
They said fuel, cash and medical supplies were running low in parts of Ukraine.
Border authorities said 29,000 people had entered Poland from Ukraine on Thursday and about half had indicated they were fleeing the war.
In Romania, more than 10,000 Ukrainians had arrived on Thursday, and nearly 3000 in Slovakia.
Poland's deputy interior minister Pawel Szefernaker said Ukrainian bus drivers were unable to drive across the border as conscription-age men were being held back in Ukraine.
Michał Mielniczuk, a spokesman for the southern Polish region of Podkarpackie said temporary accommodation was being offered to people arriving.
"The vast majority continue on to other places throughout Poland after receiving a warm meal," he told the PAP news agency.
On the border with northern Romania, women were crying as they bid goodbye to male loved ones, setting off to cross into Sighetu Marmatiei, a remote town on the banks of the Tisa river, a Reuters witness said.
Long queues had formed as cars waited to board a ferry over the Danube river into Isaccea, a town between Moldova and the Black Sea, local media in Romania showed.
Slovak authorities urged people to donate blood and set up hospitals with 5380 beds assigned for the army or NATO use.
