Long lines of cars and buses were backed up at checkpoints at the borders of Poland, Hungary, Slovakia, Romania, and non-EU member Moldova.
Others crossed the borders on foot, dragging their possessions away from the war and into the security of the EU.
Several hundred refugees were gathered at a temporary reception center in the Hungarian border village of Beregsurany awaiting transportation to transit hubs, where they could be taken further into Hungary and beyond.
Refugees fleeing from Ukraine, arrive at Nyugati station in Budapest, Hungary, on Monday, Feb. 28, 2022. AP
Maria Pavlushko, 24, an information technology project manager from Zhytomyr, a city around 100 kilometers (60 miles) west of the Ukrainian capital Kyiv, said she had been on a skiing holiday in the Carpathian mountains when she got word from home last week that Russia's invasion had begun.
``My granny called me saying there is war in the city,'' she said.
Pavlushko plans to travel from Hungary to Poland, where her mother lives. But her grandmother is still at home in Zhytomyr, she said, and her father stayed behind to join the fight against the invading Russian forces sent in by Vladimir Putin.
``I am proud about him,'' she said. ``A lot of my friends, a lot of young boys are going ... to kill (the Russian soldiers).''
Refugees fleeing conflict in Ukraine arrive at the Medyka border crossing in Poland, Monday, Feb. 28, 2022. AP
Many of the refugees at the reception center in Beregsurany, as in other border areas in Eastern Europe, are from India, Nigeria, and other African countries, and were working or studying in Ukraine when the war broke out.
Masroor Ahmed, a 22-year-old Indian medical student studying in Ternopil in western Ukraine, came with 18 other Indian students to the Hungarian border. He said they hoped to reach the capital of Budapest, where India's government has organized an evacuation flight for its citizens.
While Ternopil has not yet experienced violence in the war, he said: ``It might be that there is bombing next hour, next month or next year. We are not sure, that's why we left that city.''
Hungary, in a turnaround from its long-standing opposition to immigration and refusal to accept refugees from the Middle East, Africa, and Asia, has opened its borders to all refugees fleeing Ukraine, including third-country nationals that can prove Ukrainian residency.
As part of an agreement with some foreign governments, Hungary has set up a ``humanitarian corridor'' to escort non-Ukrainian nationals from the border to airports in the city of Debrecen and the capital, Budapest.
Priscillia Vawa Zira, a Nigerian medical student in the eastern Ukrainian city of Kharkiv, said she fled toward Hungary as the Russian military commenced an assault.
``The situation was very terrible. You had to run because explosions here and there every minute, run to the bunker,`` she said.
UN High Commissioner for Refugees Filippo Grandi tweeted on Monday that more than 500,000 refugees have now fled from Ukraine into neighboring countries.
Shabia Mantoo, a UNHCR spokeswoman, said the latest count stood at roughly 281,000 in Poland, 84,500 in Hungary, 36,400 in Moldova, 32,500 in Romania, and 30,000 in Slovakia.
The rest were scattered in unidentified other countries, she said.
A child collects toys near clothes donating point as refugees fleeing conflict in Ukraine arrive at the Medyka border crossing in Poland, Monday, Feb. 28, 2022. AP
In Poland, the country that has reported the most arrivals, trains continued to bring refugees into the border town of Przemysl on Monday.
In winter coats to protect them against near-freezing temperatures, many carried small suitcases as they lined up at the platform to exit the station.
Natalia Pivniuk, a young Ukrainian woman from the western city of Lviv, described people crowding and pushing to get on the train, which she said was ``very scary, and dangerous physically and dangerous mentally.''
``People are under stress ... and when people are scared they become egoist and forget about everything,'' she said. ``People are traumatized because they were on that train.''
Otoman Adel Abid, a student from Iraq, also fled from Lviv after he said panic broke out among many in the city.
``Everyone ran to buy some food and we heard some bombs everywhere,'' he told The Associated Press. ``After that, we directly packed our bag and clothes and some documents and we ran to the train station.''
Most of those fleeing Ukraine were entering the EU from its eastern flank, with some traveling on to countries further west.
Aksieniia Shtimmerman, 41, arrived with her four children in Berlin Monday morning after a three-day odyssey from Kyiv.
Sitting on a bench inside the German capital's main train station, she attempted to decipher a leaflet with instructions and maps on how to reach a shelter for new arrivals in the north of the city.
As she tried to comfort her crying 3-year-old twin boys, Shtimmerman said she had worked in telecommunications at a Kyiv university but was now only seeking a place where she and her children could eat, sleep and rest.
``I grabbed my kids on Friday morning at 7 a.m. to run away from the war,'' Shtimmerman said. ``I can't even count anymore how many different trains we took until we arrived here.''
Germany's interior ministry said Monday morning that 1,800 refugees from Ukraine had arrived so far, but that the number was constantly growing as more trains from Poland arrived.
In the Romanian town of Siret, the EU commissioner for home affairs, Ylva Johansson, visited a border crossing where thousands of refugees were entering from neighboring Ukraine.
Johansson, who visited some of the humanitarian stations at the border, commended the ``heartwarming'' cooperation between volunteers and the authorities, and said that the EU is united ``in a way we have never seen before.''
She said it was a ``very difficult time where we see war in Europe again, where we see aggression, invasion from Putin towards a sovereign, neighboring country.''
Johansson, who will meet later on Monday with Romanian President Klaus Iohannis and Prime Minister Nicolae Ciuca, said that Europe is ``showing that we are based on other values than Putin.''