BUDAPEST — Sending Europe’s refugee crisis hurtling toward another country, Hungary’s leaders backed down Friday from a confrontation with thousands of asylum-seekers, offering to bus the desperate migrants to the border with Austria.

The late-night offer came after days of efforts to repel the thousands of migrants fleeing war and poverty who have streamed into Hungary in a bid to reach Western Europe, where they hope to begin new lives. Hungarian Prime Minister Viktor Orban had painted his hard-line approach against the mostly Muslim asylum-seekers as a stand to preserve Europe as a Christian continent.

But after a column of migrants more than a mile long streamed onto Hungary’s main highway to Austria, it appeared that authorities felt they had no alternative but to pass the challenge to their neighbor, another country that has been ambivalent about the influx.

The Hungarian decision to provide up to 100 buses to take the asylum-seekers to the border did little to resolve the challenge facing Europe, which has failed to come up with a unified response to the mounting numbers on its borders. Instead, the plans simply shifted the crisis to another state, leaving the fundamental problem – a bloc of 503 million residents, unable to agree whether and how to house several hundred thousand refugees – to burn for another day.

“The European Union has proved to be inadequate to address the situation,” said Janos Lazar, the Hungarian prime minister’s chief of staff, in an address in parliament. He said that the decision had been made to clear the roads to ensure the country’s transportation security.

It was unclear whether the migrants would accept the offer of buses from Budapest’s main train station and from the unlit highway where many were walking in the dark. An offer a day before of trains to the border proved to be a trick to clear them into migrant processing camps.

Austrian Chancellor Werner Faymann said that he and German Chancellor Angela Merkel had agreed to take in the asylum-seekers that Hungary was moving to the border.

“On the basis of today’s emergency at the Hungarian border, Austria and Germany are allowing an onward journey of the refugees into their countries in this instance,” Faymann said in a statement on his Facebook page. But he suggested it was a one-time agreement, leaving the broader issue unresolved

In recent days Hungarian authorities had tried to halt their journey by stopping rail traffic, penning them in migrant camps and bolstering security at the border. A tense daylong standoff on a halted train about 20 miles west of Budapest ended after many of the train’s occupants stormed police lines and walked to the highway to join the larger group that had already started out from Budapest. Others – mostly women and children – agreed to go to a nearby migration processing camp.

More than a thousand asylum-seekers walked on the shoulder of the highway as trucks whizzed by Friday night. Fathers carried sleeping toddlers on their shoulders. Old men hobbled on canes. Near the front of the crowd, one man waved a blue-and-gold European Union flag.

“There’s no train, there’s no plane, there’s no taxi, there’s no one to help us, so this is the only way,” said Hala Kaman, 30, a dentist from Damascus, who had been pushing her 3-year-old son in a stroller for eight hours by late Friday. Her 10-year-old daughter and 6-year-old son walked alongside. Kaman said she hoped to go to Sweden, where authorities have promised permanent residency to Syrian refugees.

Her daughter sobbed beside her, saying she was exhausted.

“I know,” Kaman said, “but we have to keep going.”

Amid the chaos, the debate over how to respond to Europe’s refugee crisis continued to escalate. Hungarian lawmakers, fearful of the influx of asylum-seekers from conflict-torn Middle Eastern nations, approved measures Friday that gave authorities sweeping powers to seal their border and detain migrants who crossed the 108-mile razor-wire fence they have constructed across their frontier with Serbia.

“The reality is that Europe is threatened by a mass inflow of people. Many tens of millions of people could come to Europe,” Orban told Hungarian national radio Friday.