Austria said on Sunday it planned to phase out emergency measures that have allowed thousands of refugees stranded against their will for days in Hungary to stream into Austria and Germany since Saturday morning.
Many are fleeing war in the Middle East and hope to take refuge in Germany, Europe's richest country, but the EU is divided over how to cope with the influx which has provoked both huge sympathy and anti-Muslim resentment among Europeans.
Austrian Chancellor Werner Faymann said the decision, a day after the measures were put in place, followed "intensive talks" with German Chancellor Angela Merkel and a telephone call with Hungarian Prime Minister Viktor Orban.
"We have always said this is an emergency situation in which we must act quickly and humanely. We have helped more than 12,000 people in an acute situation," he said.
"Now we have to move step-by-step away from emergency measures toward normality, in conformity with the law and dignity."
Thousands of migrants and refugees arrived at Budapest's Keleti train station after traveling from Syria through the Balkans and Greece.
Hungary laid on over 100 buses to the border on Saturday night after Austria said it had agreed steps with Germany to waive the normal rules requiring refugees to apply for asylum wherever they enter the European Union.
Others set off from the station to make the 170 km (110 mile) journey on foot. The platforms filled up again on Sunday.
Germany has said it expects 800,000 refugees and migrants this year and urged EU members to open their doors. But others say the focus should be on tackling the violence in the Middle East that has caused so many to flee.
British Prime Minister David Cameron wants to hold a vote in parliament in early October to allow air strikes on Islamic State in Syria, London's Sunday Times said, and Le Monde reported that France was also considering air strikes, joining a U.S.-led coalition.
"When rich Europe argues and tears itself apart over whether to accept 1,000, 10,000, 42,000 or 100,000 refugees, when Turkey already has 2 million, it is clear that we have a problem of perspective and identity," EU foreign policy chief Federica Mogherini told Italian newspaper Corriere della Sera.
"This crisis can help us come out with a stronger vision of what it means to be the European Union."
A dozen or so well-wishers offering chocolate and bananas greeted between 600 and 700 people, mostly Syrians, arriving on two early morning trains in Munich, the state capital of Bavaria.
A total of 6,800 entered Germany on Saturday with another 5,000 expected on Sunday, Bavarian state officials said.
Merkel's decision to allow the influx has caused a rift in her conservative bloc, with her Bavarian allies accusing her of having pushed forward without asking the federal states dealing with the influx
French far-right leader Marine Le Pen accused Germany of looking to lower wages and hire "slaves" by opening its doors.
The numbers in Europe are small compared to several million refugees in Syria's neighbors Lebanon, Turkey and Jordan and Pope Francis called for every European parish and religious community to take in one migrant family each.
Volunteers drove a convoy of around 140 cars and vans filled with food and water from Vienna into Hungary to collect exhausted migrants walking to the border. Onlookers clapped and chanted: “Say it loud, say it clear, refugees are welcome here,” as they loaded their vehicles with food, water and soft toys in the Austrian capital.
But a poll in French newspaper Aujourd'hui en France showed 55 percent of French people opposed to softening rules on granting refugee status.
European leaders are due to expand their list of “safe” countries to which migrants looking for a better life can be returned. Refugees fleeing war, for instance in Syria, are expected to be given asylum.
In Hungary, migrants boarded trains at Keleti station on Sunday, following handwritten signs in Arabic directing people to trains to Hegyeshalom on the border with Austria. Volunteers handed out food and clothing.
Wrapped in blankets and sleeping bags, long lines of people, many carrying sleeping children, got off buses on the Hungarian side of the border and walked into Austria.
"We're happy. We'll go to Germany," said a Syrian man who gave his name as Mohammed.
But on Hungary's border with Serbia there were reports that people spent the night in the rain without food or shelter.
"While Europe rejoiced in happy images from Austria and Germany yesterday, refugees crossing into Hungary right now see a very different picture: riot police and a cold hard ground to sleep on,” Amnesty International researcher Barbora Cernusakova said in a statement.
Hungary, the main entry point for migrants into Europe's borderless Schengen zone, has taken a hard line, vowing to seal its southern frontier with a new, high fence by Sept. 15.
Hungarian officials have portrayed the crisis as a defense of Europe's prosperity, identity and "Christian values" against an influx of mainly Muslim migrants.
At an EU foreign ministers' meeting in Luxembourg on Saturday, the usual diplomatic conviviality unraveled as they failed to agree on any practical steps out of the crisis. Ministers are particularly at odds over proposals for country-by-country quotas to take in asylum seekers.
The flow of people risking the dangerous journey on flimsy boats across the Mediterranean shows no sign of abating, as they flee the four-year-old civil war in Syria that has killed about 250,000 civilians, and wars in Iraq, Afghanistan and in Africa.
On the Greek Island of Lesbos, about 500 Afghans protesting at lengthy identification procedures scuffled with police. A ferry took 1,744 migrants and refugees to Athens from Lesbos on Sunday and another one with 2,500 migrants was expected later in the day, the coastguard said.
A record 50,000 people hit Greek shores in July alone, and were ferried from islands unable to cope to the mainland by a government in financial crisis and keen to dispatch them into Macedonia, from where they enter Serbia and then Hungary.
More than 2,000 have died at sea so far this year. The Cypriot coastguard picked up 114 Syrian refugees who were adrift in a fishing boat on Sunday.
(Additional reporting Michael Nienaber in Berlin, Irene Preisinger in Munich; Balazs Koranyi in Budapest, Francois Murphy and Angelika Gruber in Vienna; Sybille de la Hamaide in Paris, Isla Binnie in Rome; Writing by Anna Willard; Editing by Mark Heinrich and Robin Pomeroy)