More than 7,000 Arab and Asian refugees surged into Austria and Germany following the latest U-turn by Hungary's anti-immigrant government.
The travellers, mainly from Syria, Iraq and Afghanistan, had been told for days they could not leave Hungary.
Suddenly they were taken roadsides and Budapest's central railway train station and placed on overnight buses, driven to the frontier with Austria and allowed to walk across the border.
They were met with unexpected hospitality featuring free high-speed trains, seemingly bottomless boxes of supplies, and well-wishers offering sweets and cuddly toys for the children in mothers' arms.
Even adults absorbed the sudden welcome with a look of wonderment as Germans and Austrians made clear that they had reached a land that just might become a home.
"I'm very glad to be in Germany. I hope that I find here a much better life. I want to work," said Homam Shehade, a 37-year-old Syrian shopkeeper who spent 25 days on the road.
He left behind his parents, a brother, wife, a seven-year-old boy and a two-year-old girl. He hopes to bring them all to Germany.
Until then, he said: "I hope that God protects them from the planes and bombs. My shop was bombed and my house was bombed."
As the migrants left Hungary, its leaders took a few final swipes at their departing guests.
Prime Minister Viktor Orban told reporters that Hungary drove the migrants to the border only because they were posing a public menace.
He said they had caused problems on the roads and railway tracks when they mounted a series of surprise breakouts from police-controlled positions on Friday and headed for Austria on foot.
Mr Orban said the people being taken by Germany mostly come "from regions that are not ravaged by war. They just want to live the kind of life that we have. And I understand that, but this is impossible. If we let everybody in, it's going to destroy Europe."
He said Hungary was determined to staunch the flow of foreigners traversing the country.
And he criticised EU plans to reach an agreement at a summit on September 14 committing each nation to accept higher quotas of foreigners to shelter, arguing that this would only spur more one-way traffic.
A central Budapest rally by Hungary's third-largest party, the neo-fascist Jobbik, underscored why many of those seeking sanctuary in Europe wanted to get through the country as quickly as possible.
Earlier in the week, many of the same Jobbik activists travelled south to the border with Serbia to hurl verbal abuse at newly arrived refugees.
Jobbik leader Gabor Vona told the crowd of 300 waving Hungarian and party flags: "Hungary belongs to the Hungarians. We like everybody, we respect everybody - but we don't want anybody coming here."
Other speakers branded supporters of refugee rights "traitors" and "scum".
The contrast could not have been greater in Vienna's central train station. When around 400 asylum seekers arrived on the morning's first border train, charity workers offered food, water and packages of hygiene products for men and women.
Austrian onlookers cheered the migrants' arrival, with many shouting "Welcome!" in both German and Arabic.
Sami Al Halbi, a 28-year-old vet from Hama in Syria, said he fled to avoid mandatory military service.
"They asked me to join the army. I am educated. For years I've been holding a pen. I do not want to hold a weapon," he said. "We all want to have a better future."
It got better as travellers continued west on more trains, some of them specially provided for the refugees.
As Austria's government noted, virtually none of those coming intended to seek asylum before reaching Germany, the Eurozone powerhouse that has pledged to aid Syrians fleeing from their four-year-old civil war.
Germany expects to receive a staggering 800,000 asylum seekers this year.
In Munich's central station, the first arrivals from Hungary received cheering and applause.
Many who had endured nights sleeping on crowded concrete floors at Budapest's Keleti station appeared disoriented as Germans approached them holding trays of food. The youngest brightened up as teddy bears were offered as gifts.
"We are giving a warm welcome to these people today," said Simone Hilgers, spokeswoman for Upper Bavaria government agencies tasked with providing the refugees immediate support.
"We realise it's going to be a big challenge but everybody, the authorities and ordinary citizens, are pulling together."
A total of about 6,000 people had come through Munich by Saturday evening, she said. All were given food and drink, and most were housed in temporary accommodation.
The latest arrivals add to the tens of thousands who have been streaming each month into Germany, the EU's most populous nation with 81 million residents.
The influx has strained emergency accommodation and local bureaucracy, triggered sporadic violence by neo-Nazi extremists, and inspired empathy from many more Germans.
Volunteer groups have sprung up to help asylum seekers find permanent housing and jobs, and to receive free German language lessons.