Thousands of exhausted and relieved migrants reached Austria on Saturday, stepping off a fleet of Hungarian buses to find charity workers welcoming them with open arms, offering them beds and hot tea.

The early morning move eased immediate pressure on Hungary, which is struggling to stem the flow of thousands of migrants arriving daily from non-EU member Serbia. However, officials warn that the human tide south of Hungary was still rising, and more westward-bound travelers continued to arrive in Budapest within hours of the mass evacuation of the capital’s main railway station.

Austrian police spokesman Helmut Marban told reporters that about 4,000 migrants had crossed into Austria from Hungary by mid-morning.

The country relented in its demand the travelers report to government-run asylum camps when challenged by defiant migrants largely from Syria, Iraq and Afghanistan.

Thousands marched west Friday from Keleti train station along Hungary’s main highway and camped overnight in the rain by the roadside. Hundreds more broke through police lines at a train station in the western town of Bicske, where police were trying to take them to an asylum center, and blocked the main rail line as they, too, marched west.

Austria and Germany made the breakthrough possible by announced Friday night they would take responsibility for the migrants that were already making their way west or camped out at Keleti. Hungary suspended train services from that station on Tuesday, not allowing migrants to travel westward toward Austria or Germany.

Austrian Federal Railways said the arriving migrants would be placed on trains to Vienna after passing through the border.

In jubilant scenes on the border, hundreds of migrants bearing blankets over their shoulders to provide cover from heavy rains walked off from buses and into Austria, where volunteers at a roadside Red Cross shelter offered them hot tea and handshakes of welcome. Many migrants collapsed in exhaustion on the floor, but with smiles on their faces.

The asylum-seekers in many cases have spent months in Turkish refugee camps, taken long journeys by boat, train and foot through Greece and the Balkans, then crawled under barbed wire on Hungary's southern frontier to a frosty welcome. While Austria, on Hungary's western border, says it will offer the newcomers asylum opportunities, most say they want to settle in Germany.

Since Tuesday morning, Hungarian authorities had refused to let them board trains to the west, and the migrants balked at going to processing centers, fearing they would face deportation or indefinite detention in Hungary. Government officials said they changed course because Hungary's systems were becoming overwhelmed by the sheer numbers of unwanted visitors.

Janos Lazar, chief of staff to Hungary's prime minister, said the migrants' surprise movements Friday were imperiling rail services and causing massive traffic jams. "Transportation safety can't be put at risk," he said.

The Associated Press contributed to this report.