NICKELSDORF, Austria (CNN) -- Pope Francis implored Catholic institutions throughout Europe 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 on Sunday.
"The two parishes in the Vatican these days will welcome two families of refugees."
German patience tried
Meanwhile, as more migrants poured into Hungary on Sunday, the will of the German government to relieve more pressure there -- as it had done a day earlier -- appeared to have its limits.
Germany's acceptance of thousands of people who entered Hungary while fleeing entrenched bloody conflicts in Syria, Iraq and Afghanistan should be seen as the exception and not the rule, German Foreign Minister Frank-Walter Steinmeier said on Saturday.
"The help in (Friday's) emergency situation was tied to an urgent reminder not to make that the practice for the coming days," he said at a meeting of Europe's foreign ministers in Luxembourg, according to the website of the German newspaper Die Zeit.
But on Hungary's border with Austria, a country most migrants have transited through to get to Germany, the stream of migrants continued unabated on Sunday.
By midday Sunday, Austria had let in 12,000 migrants since agreeing to allow them in a day before, according to Interior Ministry spokesman Alexander Marakovits. The border remains open to potential refugees and packed buses continue to arrive, he said.
The United Nations' refugee agency, UNHCR, estimates that 366,402 refugees and migrants have crossed the Mediterranean Sea to Europe this year, with 2,800 dead or missing. Those who make the crossing face uncertain futures in European nations, which differ in their approach to asylum seekers.
Of the thousands arriving in Austria this weekend, only a dozen or so migrants have opted to apply for asylum there, the country's Interior Ministry said. Most want to go on to Germany.
Munich police said some 5,000 refugees had arrived in the city on Sunday by 3 p.m. They told reporters that more than 7,000 entered Munich the day before.
Refugees were being registered outside the police station, where a medical tent had also been set up, spokesman Thomas Baumann said. Once registered, migrants would then be taken by bus to shelters, he said.
The country is attractive to refugees because of its robust economy, strong democracy and long history of taking in refugees. After World War II, in the face of the Cold War, Germany instituted liberal policies toward applicants for political asylum, making it a prime destination for people fleeing wars and other political strife.
Austria had stern words for other European countries over the challenge Germany has stepped up to meet.
"All of Europe must wake up now. The dreaming has to end now," said Austrian Interior Minister Johanna Mikl-Leitner on Saturday. "Now, it's up to the continent of Europe."
UNHCR spokeswoman Melissa Fleming told CNN at the Vienna train station that the situation in Austria was not sustainable and other countries needed to share the burden.
"It can't be a German problem, it's a European problem that needs to be resolved by all countries together.
"We're happy to see the outpouring of civil society in countries that are actually unwilling to accept refugees, but of course right now the refugees are saying 'We want to go to the country where we believe we'll get the best treatment, reception and can restart our lives.'"
The migrants' arrival in Austria capped an emotional week.
Most had set off on foot from Budapest, Hungary. For days, they'd been waiting at the Keleti train station, which became a focal point for the crisis. Refugees sheltering in the station refused to go to holding camps set up by the Hungarian government -- fearing mistreatment and the possibility they'd be unable to continue their journeys north.
Hungary's right-wing government wants to stop the flood of migrants. It has erected a barbed-wire fence along its more than 160-kilometer (100-mile) border with Serbia to prevent them from crossing there. On Friday, lawmakers passed bills aimed at tightening border restrictions, according to Hungarian news agency MTI. The new laws take effect next week.
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.
Warm welcome in Austria
On Saturday, under a cold rain, waves of exhausted migrants poured into the border town of Nickelsdorf, in Austria's Burgenland state.
Austrians waited at the border with Hungary to greet the convoys of buses with food and water. One group applauded and let out out welcoming cheers as weary passengers clutching children streamed across the border. The refugees carried their meager belongings in backpacks as they exited the vehicles in the rain.
The Austrian Red Cross was also on hand to provide medical supplies and warm blankets.
Search for a solution
The migrant crisis has highlighted disunity in the EU, as member countries have not been able to agree among themselves on how to handle the onslaught of migrants.
Though EU ministers meeting in Luxembourg have not yet agreed on specific policies, they did hammer out some broad general principles for dealing with the crisis: Ensure that asylum seekers are protected and treated with basic human rights; fight smugglers and traffickers; strengthen migrants' ties to their countries of origin; and work to solve the causes of the influx of refugees -- conflict in Syria and in Libya.
Addressing reporters after Saturday's meeting, EU foreign policy chief Federica Mogherini said Europe is finally starting to see the problem for what it is -- not just affecting certain EU states, such as Italy and Greece, but a Europe-wide issue.
"We also have to start using the right words. It is partially a migrant flow, but it is mainly a refugee flow, which puts us in a different situation when it comes to our legal and moral duties," she said. It's also a situation that is "here to stay," she warned.
The term refugee usually refers to people who have been persecuted politically in their homelands or who are fleeing war. The term migrant is broader and can include people crossing national borders illegally in search of better economic conditions.
Europe must remember its own history of wars and conflicts, Mogherini said, as it considers its responsibility to help and protect refugees.
UNHCR's 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."
Meanwhile, sports teams in Europe are among the nongovernmental actors offering help to the migrants.
Portuguese soccer club FC Porto on Friday appealed to all clubs in the UEFA Champions League to join it in donating 1 euro per ticket from their first home match to help resolve the humanitarian crisis.
"FC Porto is a sporting club and the social standards the club lives by mean no one is indifferent to the dramatic situation thousands of family are going through, as they often pay with their own lives for a chance at searching for the safety they lack in their home countries," it said a statement.
Soccer club Real Madrid offered $1.1 million in aid to refugees in Spain on Saturday.