Austria and Germany are expecting thousands more migrants to arrive from Hungary after Budapest eased restrictions on their travel.
Throughout Saturday, by bus, train and on foot, migrants, many of them Syrians, travelled to the Austrian border before moving on to Vienna, Munich and other German cities.
Austria and Germany are providing more trains to deal with the influx.
The German government is to discuss the crisis later on Sunday.
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.
Up to 10,000 crossed the Austrian border over a 24- to 48-hour period, according to the Austrian authorities, who have said they do not plan to limit the numbers entering.
Around 1,000 spent Saturday night in the open at Nickelsdorf on the Austrian side of the frontier waiting to be processed, officials say.
Many of those arriving in Austria travelled straight on to Munich, in southern Germany, where locals greeted them with applause, giving sweets to the children among the new arrivals.
Some 7,000 arrived at Munich station on Saturday, police say.
They have been sent on to reception centres throughout Germany to be registered and receive food and clothing.
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.
Chancellor Angela Merkel has said Germany can cope with the influx of newcomers, but there have been disagreements within her coalition government.
Some on the right say allowing so many migrants in sends the wrong signal, but the centre-left Social Democrats praised the move.
Germany is the key destination for arrivals on European shores, and expects to take in 800,000 people this year.
Syrians fleeing a brutal civil conflict are the largest group travelling, followed by Afghans and Eritreans.
Image copyright Reuters Image caption A welcome for migrants in Frankfurt early on Sunday Image copyright AFP Image caption Hungary had previously stopped migrants travelling by train to Western Europe Image copyright Reuters Image caption Austria is laying on more trains for the migrants on Sunday Image caption Shoes left for migrants by charities at the Austrian border
There is little sign of a co-ordinated EU response to the crisis, despite more than 350,000 migrants having crossed the EU’s borders in 2015 alone.
Europe’s migrant crisis is “here to stay” and nations must act together to deal with it effectively, the EU’s foreign policy chief said after “difficult” talks with foreign ministers in Luxembourg on Saturday.
“In three months time, it will be other member states under the focus, and in six months, it could be others again,” Federica Mogherini said.
Germany, backed by the European Commission, has been pushing for a quota system for sharing out newcomers between EU member states.
But this has been opposed by several eastern members.
On Saturday, Hungary said that while it had temporarily relaxed restrictions on the transit of asylum seekers, it was pressing ahead with plans to tighten border controls and could send troops to its southern frontier if parliament agreed.
A border fence is due to completed by 15 September.