A mother and her child sit in a bus, which is supposed to leave to Austria and Germany, at the Keleti trainstation in Budapest, Hungary, September 4, 2015. [Photo/Agencies]

HEGYESHALOM, Hungary - Hundreds of exhausted migrants streamed into Austria on Saturday, reaching the border on buses provided by an overwhelmed Hungarian government that gave up trying to hold back crowds that had set out on foot for western Europe.

After days of confrontation and chaos, Hungary's right-wing government deployed dozens of buses to move on migrants from the capital, Budapest, and pick up over 1,000 - many of them refugees from the Syrian war - walking down the main highway to Vienna.

Austria said it had agreed with Germany that they would allow the migrants access, unable to enforce the rules of a European asylum system brought to breaking point by the continent's worst refugee crisis since the Yugoslav wars of the 1990s.

Wrapped in blankets against the rain, hundreds of visibly exhausted migrants, many carrying small children, climbed off buses on the Hungarian side of the border and walked in a long line into Austria, receiving fruit and water from aid workers.

"We're happy. We'll go to Germany," said a Syrian man who gave his name as Mohammed.