By Peter Murphy with Bryan McManus
Thousands of migrants have streamed into Austria, desperate to travel on to Germany after days of ugly standoffs with authorities in Budapest's train station that made headlines worldwide.
Austria's interior ministry said 6500 people had crossed into the country since Friday night when Hungary laid on more than 120 buses to ferry the migrants to the Austrian border.
But on Saturday (local time), Hungary's police chief announced the end of the bus transfer, calling it a "one-off."
Columns of refugees set out on foot to walk the 175km to the German border.
Hungary's tough line contrasted with a show of solidarity elsewhere in Europe, reflecting divisions over how to handle the continent's biggest refugee crisis since the end of World War II.
"This has to be an eye opener, how messed-up the situation in Europe is now," Austrian Foreign Minister Sebastian Kurz said on arriving in Luxembourg for European Union (EU) talks dominated by the crisis.
"I hope that this serves as a wake-up call."
Finnish Prime Minister Juha Sipila offered to put up refugee families in his country home, in France more than 10,000 people rallied in support of the migrants and in Frankfurt, hundreds of Germans gathered at the city's railway station, ready to welcome refugees with water, food and clothing.
Around 4300 migrants boarded trains in Salzburg for Germany on Saturday, with no incidents reported, Austrian officials said.
Germany's open policy on asylum has made it a magnet, especially for those fleeing war-ravaged Syria.
The bill for the German government is expected to be around 10 billion euros ($A15.9 billion) this year, according to the Frankfurter Allgemeine newspaper.
But Chancellor Angela Merkel insisted Berlin could still balance its budget and fulfill its "duty" as a haven.
At the Austria-Hungary border, Red Cross medics were on hand to tend to the sick and injured.
"We treated a two-day-old gunshot wound. We're seeing eye injuries caused by stun grenades. We're seeing bruising, including children with bruising," Red Cross spokesman Andreas Zenker said.
Most boarded special buses and trains to Vienna, from where they planned to continue on to Munich, or other German cities.
The perils of thei migrants' odyssey could be seen on Saturday on the Greek island of Agathonisis where a two-month-old baby died shortly after the family landed there from Turkey. Authorities did not immediately provide the cause of death.
It comes just fews days after a three-year-old Syrian boy, Aylan Kurdi, washed up on a beach in Turkey, the photographs vividly showing the horrific human cost of the Syrian exodus.
Aylan, his brother and mother died after the boat in which they were crossing the Aegean Sea to Greece capsized.
On the Greek island of Lesbos, riot police used tear gas Saturday night to drive back thousands of migrants who had tried to force through barriers to board a ship chartered to transfer some of them to Athens, police said.
EU foreign ministers, meeting in Luxembourg, discussed proposals for mandatory quotas to resettle 160,000 refugees across member states.
But the 28-nation bloc is divided roughly along east-west lines, with relative newcomers to the EU from the former Soviet bloc taking a harder stance.
"We must not forget that those who are coming in have been brought up under a different religion and represent a profoundly different culture," Hungarian Prime Minister Viktor Orban wrote in a German newspaper this week.
"The majority are not Christians but Muslims. That is an important question because Europe and European culture have Christian roots."