MUNICH — Germans waving welcome signs in German, English, and Arabic came to the train station here Saturday to greet the first group of what is expected to be about 8,000 migrants to arrive in Germany by early Sunday, after an arduous and emotional journey through Hungary and Austria.

Germans applauded and volunteers offered hot tea, food, and toys as about 450 migrants arrived on a special train service from Austria, finally reaching Germany, which had held out an open hand to them.

“Thank you, Germany,” said one woman from the Kurdish part of northern Iraq who said she had been on the road for a month and a half with her two children.

A German volunteer, Silvia Reinschmiedt, who runs a local school, could not stay at home. “I said to myself, I have to do something,” she said as she handed out warm drinks.

By Saturday evening, about 6,000 migrants had arrived here, and another 1,800 were expected to arrive in trains overnight, according to the German police.

It was the desired destination for an extraordinary march of migrants, who broke through Hungarian obstacles and reached Austria on Saturday morning after a night of frantic negotiations among German, Austrian, and Hungarian officials cleared the way.

Overnight, some 4,500 exhausted migrants were bused to the Austrian border by a Hungarian government that gave up trying to stop them and instead decided to help them travel in safety. That help was temporary, however, as Hungary found itself struggling to cope with a new influx of migrants.

The arrival in Germany of the migrants was the culmination of 10 days of tragedy and emotion that at last caught the world’s attention, as war and chaos in Syria and elsewhere in the Middle East set off one of the largest emigrations since World War II.

The standoff in Hungary seemed to encapsulate the long and often deadly journeys that hundreds of thousands of people have made to try to reach some semblance of peace, security, and prosperity in a Europe that, for the most part, did not much want them.

Even as the thousands made it to Austria on buses provided by the Hungarian government, on Saturday morning a new group of about 1,000 migrants set off on foot from the Budapest train station, Keleti, on their own march to the border.

At the same time, at least 2,000 more migrants were caught trying to enter Hungary on Friday alone, and Janos Lazar, chief of staff to Prime Minister Viktor Orban, said that Hungary would work to complete its border fence to stop further illegal entry.

Zoltan Kovacs, a government spokesman, told the state news agency that Budapest was not planning to send any more buses to Austria. The Hungarian authorities, worried that easing the migrants’ journey would just encourage more to attempt the passage, said Saturday that they would stick to their understanding of European regulations and try to stop and register new migrants, again leaving thousands stranded.

The drama highlighted some serious policy questions for European foreign ministers meeting in Luxembourg on Saturday. How many migrants would be welcome and for how long? How much has Germany’s “open door” encouraged more migrants to embark on the often-treacherous journey?

The meeting of the foreign ministers produced little agreement; the bloc’s foreign policy chief, Federica Mogherini, described the talks as “difficult.” Europe’s migrant crisis is “here to stay,” she said, and nations must act together. “In three months’ time, it will be other member states under the focus, and in six months, it could be again others,” she said.

The European Union, which operates by consensus among its 28 member states, is debating what to do, but considerable resistance remains among central European states like Poland, Slovakia, and the Czech Republic, as well as from Britain, to accepting mandatory quotas of migrants, as France and Germany have proposed.

According to figures from the UN refugee agency, about 49 percent of the 320,000 or so migrants who have reached Europe this year are from Syria and only 3 percent from Iraq. About 12 percent are from Afghanistan and 8 percent from Eritrea.

In 2014, only about 45 percent of asylum applications made to European governments had a positive outcome — at least half were turned away for not being legal refugees but illegal migrants.

The EU bureaucracy is trying to come up with a plan to set up reception centers for migrants in Greece and Italy, where they can be cared for and screened. The officials are also drawing up a plan to distribute up to 160,000 migrants and asylum seekers. But the countries must agree.

European interior ministers will meet Sept. 14 to discuss the proposals and a summit meeting of bloc leaders is likely to follow — unless one is called sooner under the pressure of events.

“This has to be an eye-opener on how messed up the situation in Europe is now,” said the Austrian foreign minister, Sebastian Kurz. “I hope that this serves as a wake-up call that this cannot continue.”