Live Updates: Zelensky Posts Video From Kyiv as Ukraine’s Forces Stall Russian Advance

In the evening, she got a call: A family from Lviv was on their way.

“We didn’t even discuss it,” said Ms. Hawranek, a freelance journalist whose partner, also a journalist, immediately left to cover the war. “It was obvious that we were going to do this.”

Of the 1.7 million people who have fled Ukraine since the beginning of the invasion, more than one million have made their way to Poland, according to the United Nations.

This huge, sudden influx of refugees has given rise to an enormous grassroots movement across Polish society, as private individuals have mobilized to raise funds and offer free accommodation and transport to the refugees.

More than 500,000 Poles have joined a nationwide Facebook group coordinating support. In some places, supply was greater than demand, with local authorities calling on citizens to refrain from driving to the border to offer free rides, because they were causing traffic jams.

Years of nationalist, anti-refugee policies have left Poland with a fragmented immigration system. It’s now mostly up to citizens to handle what the U.N.H.C.R. said was “the fastest growing refugee crisis in Europe since World War II.”

Ms. Hawranek’s guests arrived on Friday night: Kostiantyn Komkov, a software developer, Olena Poretskova, a costume designer, and their 5-year old son, Tomas. As soon as the invasion started, the family immediately left their Lviv apartment to friends who were evacuating from Kyiv, and crossed the border to Poland. “I had anticipated an attack for the past two years, and when I saw the Russian troops building at the border, I knew this was it,” Ms Poretskova said.

For Tanja Fedchyk, a nurse from Luck in western Ukraine, who has also found asylum in Poland, the decision on whether to stay or go was not instantaneous. When the Russian army first entered eastern Ukraine, she and her husband decided to wait 24 hours. “We were hoping that the situation wouldn’t develop into a full-scale invasion,” Ms. Fedchyk said. “But as hours passed, it became obvious that things were only getting worse.”

The next morning, Ms. Fedchyk and her 2-year-old son, Tymi, got in a car and headed for Wroclaw, Poland. The trip went relatively smoothly, except for a 10-hour wait at the border. But saying goodbye to their husband and father, who stayed in Luck to build barricades, left them heartbroken.

In Wroclaw, they are hosted by Robert and Hana Reisigová-Kielawski, an English language university instructor and a human-resources supervisor, who live with their two children. The couple didn’t have a spare room in the apartment so they moved their 5-year-old daughter to their bedroom.

“As we waited for their arrival, we got nervous,” Mr. Reisigová-Kielawski said. “We had no idea what physical and emotional state they would be in. I wondered how we should behave in order to be as helpful as possible, but also not overwhelm them. Which issues should we discuss and which are best left unsaid?”

One thing was clear from the beginning: They wouldn’t ask their guests how long they were planning to stay. Their invitation didn’t have an expiration date.

But whenever they asked if Ms. Fedchyk needed anything, she would say, “No, thank you. We’re just here for a few days.” As the invasion unfolded, however, it became evident that those days could turn into weeks, perhaps longer.

Since the war began, Ukrainians on both sides of the border have faced uncertainty. In Poland, the government is preparing an emergency bill that will make it easier for Ukrainians to access the labor market and some of the social benefits available to permanent residents.

Commentators have pointed out that the warm welcome Ukrainian refugees have received stands in stark contrast to the public response to the humanitarian crisis at the border with Belarus, which peaked in October. The government did not open the border to those refugees, most from the Middle East, and it banned aid workers from the border region — policies widely supported by Poles.

The Reisigová-Kielawskis, long active in various refugee-support programs, were frustrated.

“During that crisis the government made it extremely difficult for Poles to help refugees, and unfortunately many people chose to look away,” Mr. Reisigová-Kielawski said, adding. “The grassroots movement to help Ukrainians, which we are seeing at the moment, is immense and heartwarming, but I have the impression that it is also lined with a sense of guilt that as a society we didn’t do enough back then.”