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Exodus Grows as Nearly 370,000 People Flee Russia’s Ukraine Invasion. Millions May Follow

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Nearly 370,000 people have fled Ukraine since Russia launched an unprovoked military invasion last week, the United Nations High Commissioner for Refugees said Sunday, an exodus that the U.N. estimates could grow to as many as 5 million. The U.N.’s refugee agency said half of those fleeing had headed to the 300-mile border with Poland, where U.S. forces are helping with preparations for new arrivals, while other displaced civilians left for Hungary, Moldova and Romania.

The refugee numbers are expected to intensify in the coming days as Russian ground forces push toward major cities, including the capital Kyiv with its population of 3 million people. As the sun set on Saturday night, missiles rained down on population centers while bursts of gunfire could be heard amid intense street battles. Ukrainians crowded onto rail cars, roadways and even opted to flee on foot, sparking what many American and European officials fear is the next refugee crisis on the continent.

“If Russia continues down this path, it could–according to our estimates–create a new refugee crisis, one of the largest facing the world today, with as many as 5 million more people displaced by Russia’s war of choice and putting pressure on Ukraine’s neighbors,” Linda Thomas-Greenfield, U.S. ambassador to the U.N., said this week.

The Russian military advance is going slower than U.S. intelligence anticipated, a senior U.S. defense official said Saturday, with Ukrainian forces showing stiff resistance. No major Ukrainian city has yet fallen to Moscow. Airspace over Ukraine remains contested, despite predictions that Russian fighter jets would quickly dominate the skies.

Still, Russia has launched more than 250 missiles into Ukraine, many of which slammed into civilian areas, reportedly blowing apart homes, hospitals and apartment buildings. The bombardment has already damaged critical water infrastructure and education facilities in recent days, according to the U.N. Most of those fleeing are women, children and the elderly following Ukrainian President Volodymyr Zelensky’s order banning men aged 18 to 60 from leaving the country.

Beyond those Ukrainians streaming across borders to the west and south, tens of thousands of other families are expected to be displaced within the country, potentially fueling a humanitarian crisis.

Read More: ‘We Will Defend Ourselves.’ Photographs of Ukraine Under Attack

Poland announced its border will be open to fleeing Ukrainians, even those that do not have official documents. In Poland, U.S. paratroopers with the 82nd Airborne Division have helped prepare facilities to process refugees. Reception points have been set up for those seeking meals, medical assistance and information. “We do continue to see increases in the numbers of people trying to leave the country,” a senior U.S. defense official said. “The lines are stacking up on the Ukrainian side of the border with Poland.”

Columns of would-be refugees stretched for miles along roads leading from Ukraine into Poland. Many of those fleeing had minimal possessions, taking what they could before leaving their homes. The mass migration has the potential to surpass the 1.3 million asylum seekers who fled conflicts in Syria, Iraq, Afghanistan and sub-Saharan Africa in 2015. That exodus was Europe’s largest wave of refugees since World War II and helped fuel the rise of populist far-right parties and anti-immigration sentiment.

Poland’s position on the incoming refugees is a dramatic turn from last year when baton-wielding Polish security forces repelled families who made the treacherous journey from war-torn states including Afghanistan, Iraq and Yemen. Amnesty International spoke to people who suffered violent treatment at the hands of Polish authorities during that crisis, including children being pepper sprayed and adults being pushed into rivers.

About 2 million Ukrainians already call Poland home, many having fled their homes following Russia’s illegal seizure of the Black Sea peninsula of Crimea in 2014 and its orchestration of an eight-year insurgency in eastern Ukraine. Polish Interior Minister Mariusz Kaminski said Thursday that his country was prepared for another wave. “We will do everything to provide safe shelter in Poland for everyone who needs it,” he said.

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Write to W.J. Hennigan at william.hennigan@time.com