Here's a look at the latest country-by-country developments in the refugee and migrant crisis unfolding across much of Europe:
Around 11,000 migrants and refugees have crossed into Austria from Hungary since Saturday, Austrian Interior Ministry Spokesman Alexander Marakovits told CNN on Sunday. The Austrian border remains open and more migrants are arriving in buses from Hungary, Marakovits said.
Austrians waiting at the border cheered as refugees arrived, left the buses and crossed the border on foot -- welcoming them with water and food.
The Interior Ministry said Saturday that only a dozen or so of the thousands of refugees had applied for asylum in Austria. The rest had continued on to Germany.
Austria earlier 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 Saturday. "Now, it's up to the continent of Europe."
On Friday, Interior Ministry spokesman Karl Heinz Grundboeck in Vienna said his country was not planning to change policies along its border with Hungary in light of the massive influx of migrants across Central Europe. Austrian officials don't currently enforce border controls on their side of the border under terms of a 1985 agreement that eliminates such internal controls in 25 European nations. Grundboeck could not comment on what Hungarian police might or might not do on their side of the border.
Hungary: More migrants pour in
Many of the migrants crossing into Austria had been transported by bus after earlier walking for hours out of Hungary's capital, Budapest.
The mass march began Friday when more than 1,000 men, women and children trudged along the side of a highway outside Budapest, carrying their belongings with them, CNN's Arwa Damon said at the scene.
People in the throng told CNN they had decided to set out on foot because they simply could not wait in the Keleti train station any longer for Europe's politicians to make decisions -- and that they'd walk all the way to Germany if they had to. The Austrian capital, Vienna, is about 250 kilometers (150 miles) from Budapest.
The government sent 100 buses to transport to Austria the people walking along the roadway, as well as people at Budapest's main train station, spokesman Zoltan Kovacs said.
But migrants have been wary of Hungarian authorities' transport offers, fearing they want to place them in holding camps -- an option they reject because they say they were badly treated at camps when they first crossed into Hungary from Serbia, which Hungary denies.
Hungary's right-wing government is building a barbed wire fence along the Serbian border. And on Friday, lawmakers passed a package of bills -- due to take effect next week -- aimed at tightening border restrictions to prevent migrants from entering illegally, Hungarian news agency MTI reported.
Germany: Acceptance an exception -- not the rule
As more migrants poured into Hungary on Sunday, the will of the German government to relieve more pressure there 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.
Germany 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.
European Union: Europe-wide issue
EU foreign ministers began an informal gathering in Luxembourg on Friday to discuss the crisis.
Addressing reporters Saturday, 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.
Mandatory quotas have been rejected by some EU members. However, a voluntary system makes decisions more difficult, she said, so common systems are needed to speed up the process.
More must also be done to tackle the networks of human traffickers exploiting migrants who seek to cross the Mediterranean from North Africa to Italy, or to travel from Greece via the Balkans to northern Europe, she said. Restoring stability to Libya and finding a political solution to the war in Syria are key to resolving the issue longer term, she said.
France: Moral obligation
France and Germany have proposed a welcoming mechanism that would be permanent and mandatory in Europe, French President Francois Hollande said Thursday.
"What exists today is no longer enough, and there are countries -- and I will not name them here because we have to work with all of them -- but who are not responding to their moral obligations," Hollande told reporters.
"Europe is a group of principles, of values which oblige us to welcome those who are pushed out and look for refuge because they are persecuted."
Finland: Prime Minister offers home
Finland's Prime Minister has offered to open his house to asylum seekers in need of accommodation, the country's state broadcaster said Saturday.
Prime Minister Juha Sipila promised to accommodate asylum-seekers coming to Finland in his home in Kempele, near the coastal city of Oulu, saying he wants to encourage all Finns to help with the refugee crisis, Finnish broadcaster YLE said.
The Prime Minister has also appealed to churches and volunteer organizations to help with the growing need for accommodations. "I hope this becomes some kind of people's movement that will inspire many to shoulder part of the burden in this refugee housing crisis," Sipila said. "What we need now is a show of compassion."
Greece: Thousands more arrive each day
More than 5,000 migrants and refugees per day have crossed the Aegean Sea into Greece over the past week, the International Organization for Migration said Friday.
The largest group is Syrians, followed by Afghans, it said.
Many of the families, especially the Afghans, include pregnant women and newborns, the IOM said.
The small islands of Kos and Lesbos have found themselves on the front lines of the crisis. Many new arrivals are having to wait several days to be registered by police, camping out by overstretched reception centers.
Rights group Amnesty International said its team in Kos had witnessed a violent attack Thursday night on refugees by a group of 15 to 25 people brandishing bats and shouting "Go back to your countries" and other abuse. Riot police intervened with tear gas only after the physical attacks had started, the rights group said.
United Kingdom: Pressure on to take more refugees
The United Kingdom "will accept thousands more" Syrian refugees, Prime Minister David Cameron said Friday.
"Britain will act with our head and our heart, providing refuge for those in need while working on a long-term solution to the Syria
crisis," he said.
Cameron insisted Britain was doing its part, saying it had already accepted about 5,000 Syrian refugees, but would now take in more vulnerable Syrians from camps in neighboring countries.
"As the second-largest bilateral donor to the crisis, we have provided over 900 million (British) pounds in aid to help those affected in Syria and the region. We have funded shelter, food, water and vital medical supplies for millions of desperate refugees fleeing the conflict and helping them to survive in the countries around Syria, like Jordan and Lebanon," he said.
United Nations: Piecemeal approach doesn't work
On Friday, the head of the United Nations refugee agency called for a common approach to tackle what he said was "a primarily refugee crisis, not only a migration phenomenon."
"Europe cannot go on responding to this crisis with a piecemeal or incremental approach. No country can do it alone, and no country can refuse to do its part," U.N. High Commissioner for Refugees Antonio Guterres said in a statement.
A new system must be set up, he said, based on a common policy.
"Concretely, this means taking urgent and courageous measures to stabilize the situation and then finding a way to truly share responsibility in the mid to longer term."