A convoy of cars driven by German and Austrian activists has crossed into Hungary to pick up migrants and help them reach western Europe.
Thousands – many of whom initially fled conflict in Syria – have made their way through Austria since Hungary removed restrictions on transit on Friday.
Buses and special trains have been taking them from the Hungarian border to Vienna and on to Germany.
The Pope said every Catholic parish in Europe should host a migrant family.
Speaking during the Angelus prayer, Pope Francis appealed for “every parish, every religious community, every monastery, every sanctuary in Europe” to take in a family.
One of the Austrian activists taking part, Angelika Neuwirth, told the BBC that their aim was to take them back to shelters in Austrian capital.
“I think this is my duty. I’m a mum, I’m a woman from Austria and I can’t close my eyes anymore,” she said. “We are all human. No-one is illegal.”
A key stage in the current crisis was reached on Saturday, when up to 10,000 migrants travelled by bus, train and on foot, to the Austrian border before moving on to Vienna, Munich and other German cities.
After days of confrontation and chaos, Hungary opened its borders with Austria and bussed thousands of migrants to the frontier. Many, frustrated at being prevented from boarding trains in Budapest, had begun to walk along a motorway towards Austria.
The first arrivals in Munich were greeted with applause, and children were handed sweets. Since Saturday morning, 8,000 migrants have arrived at Munich station.
Many of the migrants had travelled north through the Balkans – Greece, Macedonia and Serbia – before arriving at Hungary’s southern border.
Three-thousand arrived in Presevo on the Serbian side of the frontier with Macedonia on Saturday, most spending the night in tents or in the open, reports say.
Just inside Hungary, between 200 and 300 migrants broke out of a processing centre in Roszke, demanding to be allowed to proceed to Germany, Hungarian media reports.
Both Germany and Hungary have said the current measures are aimed at averting a humanitarian crisis, and will not set a precedent.
The rules requiring refugees to apply for asylum in the first country they land in “are still valid, and we expect other European Union member states to stick to them”, a German government spokesman said.
However, in August Germany waived European Union rules on asylum seekers from Syria, allowing them to register in Germany regardless of where they first entered the EU.
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