VIENNA -- Thousands of migrants, most of them from war-torn Syria, finally crossed from Hungary into Austria on Saturday.

Hungary has been overwhelmed by the flood of migrants escaping violence and poverty in the Middle East and North Africa who are trying to begin new lives in Europe.

In the Austrian town of Nickelsdorf, not far from the border, the migrants arrived in a place a world away from where they've just come. The welcome was as comforting as the helpings of the rice being served up.

Austrian police said 6,000 migrants reached the country Saturday alone.

They'd been stuck in Hungary for days or weeks, and many ran out of food, water and patience long ago.

"Now we are free," one man said. "For five days in Hungary, we were in a very, very bad position. In a very, very bad position. And Allah, thanks God, thanks God."

Matina Adelsberger said she saw what was happening to the migrants and had to help.

"I'm so happy they could be here, and we could give them something to eat, and that they see they are welcome here," Adelsberger told CBS News.

A welcome that included warm clothes for Kisam Zukari's daughters, who haven't had much reason to smile since they fled the Syrian city of Latakia.

Among the volunteers: Syrian refugee Lohi al Hussein, who made that same perilous journey himself a few months ago.

"Austria is very nice, very good country," al Hussein said. "I like it, really. I want to stay here."

But not everyone can get here.

Hungarian officials announced that the fleet of buses that brought migrants to the border would not be provided again.

That news triggered another mass exodus, following in the footsteps of those thousands who marched toward the Austrian border on Friday.

The target destination for most of these people is Munich, Germany, where migrants arrived to applause and more help.

But many more migrants have just arrived in Hungary. Until the country comes up with a system that works, their future is far from certain.

Austria's government said Saturday that, unlike Hungary, they won't stop migrants who want to head to Germany.

Other countries were also prepared to take in more people. Norway was expecting as many 16,000 refugees.

Ireland and the U.K. have promised to take in nearly 6,000. Kosovo, Europe's poorest country, will take in 3,000.

Finland is expecting to receive 30,000 refugees -- the country's prime minister even offered up his unused home.

Germany by far remains the most generous of foster countries, setting no limit on the number of people who could go there.