After absorbing more than 12,000 refugees, Austria wants to see a gradual reduction in the numbers of refugees coming through, Austrian Chancellor Werner Faymann said.
"We have always said this is an emergency situation, which we have to handle quickly and humanely," Faymann said Sunday. "We have helped more than 12,000 people in an acute situation. We must now step by step go from emergency measures to a normality that is humane and complies with the law."
Austria's border with Hungary remains open to potential refugees, Austrian Interior Ministry spokesman Alexander Marakovits said Sunday, and packed buses and trains continued to arrive.
Despite the government's desire to curb migrant flow, there has been an outpouring of support from Austrians. Many have brought food and water and cheered for the refugees pouring onto the platform at Vienna's train station.
The Austrian Red Cross has also been on hand to provide medical supplies and warm blankets.
One recent arrival, standing with his two daughters, told CNN of the family's difficult journey through Hungary.
"We went through a torture," he said. "We walked 110 kilometers (68 miles) with the children. They didn't allow us to take cars or trains."
But the Hungarian people, he said, "were very nice. We arrived here safely, and we are comfortable here, and we like the people and the government of Austria."
Most of the arrivals in Austria intend to travel farther into Europe, however. Of the thousands who have arrived there this weekend, only a dozen or so have opted to apply for asylum there, the country's Interior Ministry said.
German patience tried
Germany is attractive to refugees because of its robust economy, strong democracy and long history of taking in refugees. But it, too, can't keep taking in refugees at the current pace.
"The great helpfulness that Germany has shown in these last weeks and months should not be worn thin," the Interior Ministry said in a statement Sunday.
It called for all European countries to work together and share responsibility.
"Only if that is guaranteed can Germany continue its do its part in helping the large number of asylum-seekers," the statement said.
Foreign Minister Frank-Walter Steinmeier said as much Saturday, telling his European counterparts that Germany's acceptance of the thousands fleeing conflicts in Iraq, Syria and Afghanistan must not be the practice in the coming days, according to the website of the German newspaper Die Zeit.
The government has said it will accept 800,000 applications for asylum from refugees. A poll released by broadcaster ARD on Thursday found 88% of Germans willing to donate clothes or money to refugees, and 67% willing to volunteer to help them.
Some 5,000 migrants arrived on Sunday at the train station in Munich, according to police there. The country's interior ministry said around 8,000 arrived in southern Germany the day before.
After arriving, the people are registered, fingerprinted and brought to shelters or temporary housing such as university dorms or even shipping containers that have been converted into living spaces.
Pope implores European Catholics to help
Pope Francis implored Catholic institutions throughout Europe on Sunday to show mercy to the flood of refugees arriving on their shores by offering them shelter.
"May every parish, every religious community, every monastery, every sanctuary of Europe host a family, starting from my diocese of Rome," Francis said at the end of his Angelus prayers in Rome.
"The two parishes in the Vatican these days will welcome two families of refugees."
The Pope's urging came as more than 12,000 migrants poured into Austria on Sunday, the Interior Ministry said, arriving at train stations to applause, cheers and pats on the back from Austrians eager to help them.
There were similar scenes at train stations in Germany. In Saalfeld, a crowd of locals stood on the platform clapping and singing in English, "Say it loud, say it clear, refugees are welcome here" as migrants arrived.
Hungary's treatment 'absolutely unacceptable'
Many of the migrants arrive with harrowing tales of crossing the Mediterranean, then walking from Greece through Macedonia, Serbia, Hungary and finally into Austria.
It is in Hungary that many say they endured the worst treatment. Buses took them to two transit camps, where migrants and their advocates say conditions are inhumane. CNN was not allowed in.
Amnesty International representative Barbora Cernusakova said she visited the camp in Roszke, just over the border from Serbia, on Sunday.
"There were a number of refugees who didn't want to stay at this collection point, and I don't blame them because there are absolutely no facilities," she told CNN.
She said migrants have complained of overcrowding, no access to toilets and sanitation facilities, and verbal abuse by police.
"These are simply not conditions where you put a person who's been through a traumatizing journey through several countries," she said. Many of these people have been walking for a number of days, and actually these people are in many, many cases escaping an armed conflict. This is absolutely unacceptable."
Hungary's right-wing government, trying to stop the flood of migrants, has erected a barbed-wire fence along its more than 160-kilometer (100-mile) border with Serbia to prevent them from crossing there. Lawmakers last week passed bills aimed at tightening border restrictions, according to Hungarian news agency MTI.
Hungary's government has blamed the onslaught of migration on the European Union's immigration policy and on statements made by European politicians that could be understood as welcoming migrants.
UNHCR spokeswoman Melissa Fleming said there are a number of things European nations could do to help.
"First of all, once people arrive in Europe, what we're proposing is that there be reception centers, registration centers run by the EU, supported by UNHCR in Greece and Italy and also in Hungary so that they could claim asylum in those countries," she said.
"But it would only work if there is a relocation in place into other countries of Europe -- not just Germany, not just Sweden, not just Austria."