By Justin Spike and Vanessa Gera | Associated Press

TISZABECS, Hungary (AP) — Of the hundreds of refugees gathered on the grounds of a small village school in eastern Hungary, almost all are women and children who left their husbands, fathers, brothers and sons behind to fight in Ukraine’s resistance to the deadly Russian invasion.

“I have brothers, they are fighting now,” said Olga Skliarova, a 34-year-old resident of Ukraine’s capital, Kyiv. “Men are not allowed to cross the border, so they helped us to get to the border and went back to Kyiv to fight.”

The exodus of refugees from the war in Ukraine is rapidly growing in the eastern countries of the European Union, with more than 675,000 people fleeing to neighboring countries since the Russian invasion began — a number that will only grow, according to the U.N. refugee agency.

Shabia Mantoo, a spokeswoman for the U.N. High Commissioner for Refugees, said in Geneva Tuesday that “at this rate, the situation looks set to become Europe’s largest refugee crisis this century.”

An order from Ukraine’s government prohibiting men aged 18- to 60-years-old from leaving the country — so as to keep them available for military conscription — means that many women and children must seek safety on their own.

Irina Yarimchuk, an accountant from the western Ukrainian town of Kalush, traveled the five hours to the Hungarian village of Tiszabecs early Tuesday with her 14-year-old son and 1-year-old daughter.

Through tears, she said her brother had joined the Ukrainian army, and she was “very worried about his life.”

“I love you so much. Keep yourself strong. We will win, and we will see you soon, I hope,” she said in an emotional message to her brother, who is stationed near Ukraine’s border with Belarus.

After a missile hit the nearby Ivano-Frankivsk airport as the Russian invasion began Thursday, Yarimchuk — who is planning to stay with relatives in Prague — spent her days and nights shuttling her family from their home to a bomb shelter as air raid sirens blared every few hours.

“From that day … we stay outside our home every evening,” she said of the five panicked nights before she left. “I was afraid for my children.”

Skliarova, who worked in tourism in Kyiv before war laid siege to the city, recounted keeping a full backpack of clothes and emergency supplies that she brought with her each time she took shelter in her apartment building basement.

“It’s an evil, evil feeling,” she said. “Scary, stress, shock. Every hour in the night, we got up to run into the basement. We slept dressed to get up and run.”

In Poland, too, it was largely Ukrainian women arriving with their children as Putin’s forces intensified their attacks on civilian targets in a campaign that is becoming deadlier for children.

Ukrainian President Volodymyr Zelenskyy told the European Parliament on Tuesday that Russian forces had killed 16 children the day before as he appealed to EU leaders to accept Ukraine in the bloc.

Among those fleeing was Oxana Sereduk, who arrived by car in Medyka, Poland, with her two daughters and grandchildren Tuesday morning. Her daughter Mariana drove the car, sometimes with her 16-month-old baby sitting on her lap and nursing.

“I am mostly afraid for the children,” Sereduk said.

Maria Lisicka took her two children and fled when shelling began in Lutsk, western Ukraine. “I will do everything for my children,” she said. “I didn’t want to take them away, I wanted them to be at home, but what can be done? I want their psyche to be normal. The most important thing is children. I don’t care about the rest.”

At the bus station in the western Ukrainian city of Lviv on Tuesday, Pavlo Bilodid wiped away tears as he kissed his wife and 2-year-old daughter Maria goodbye, helping them board the bus to Poland.

Wrought with emotion, as three generations of the female members of his family made the trip to safety, he told the Associated Press they had fled Kyiv after fighting there intensified.

“Sorry,” he said, taking a moment to compose himself. “It’s terrible, because it was so unexpected and nobody was prepared for this situation, and we believe that we will see them soon. I’m sure that we will see them soon.”

Bilodid, 33, and his father stayed behind.

“We’re staying here and we have to volunteer and we will do here what we can do here in Lviv, and if we will need, we will go to Kyiv to fight,” he said.

Back in Hungary, many of the women and children at the school in Tiszabecs hope to reach destinations in the Czech Republic and Poland, but difficulties have arisen in arranging transport from the remote village on the Ukrainian border.

“Our biggest lack in recent days has been transportation,” said Lajos Revesz of Hungarian Baptist Aid, which is running the reception center at the school.

Volunteers from several countries in the region have begun arriving in Tiszabecs to bring newly-arrived refugees farther into the EU.

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Ivan Mursha, a native of Khust in western Ukraine who lives in Brno, Czech Republic, drove more than seven hours to offer free rides to anyone wishing to join him.

“We are all Ukrainians. We must unite and help each other,” Mursha said.

Skliarova, before departing for Brno in Mursha’s 12-passenger van, said that her cousin’s 15-year-old son was shot by Russian forces on Monday while riding in a car in Brovary, on the outskirts of Kyiv. The boy, she said, survived.

When asked what she expects to find when she returns to Kyiv, she said simply: “Ruins.”

“I think someone has to stop” Vladimir Putin, she said. “I think there must be at least one person in the world who can stop this paranoiac. He’s ill.”

Gera reported from Medyka, Poland. AP writer Andrea Rosa in Lviv, Ukraine, contributed to this report.