Refugees wait to cross the Greece-Macedonia border on Sept. 10. (Sakis Mitrolidis/AFP/Getty Images)

Thousands more refugees filled choke points across Europe on Thursday as overwhelmed leaders tried to rally support for a resettlement plan that even its backers acknowledge is not enough to handle the crush.

“One could also say a drop in the ocean that won’t solve everything,” German Vice Chancellor Sigmar Gabriel told Parliament as he outlined the proposal to spread 160,000 refugees across nearly two dozen countries.

In Washington, President Obama has directed the government to prepare to accept at least 10,000 Syrian refugees next fiscal year, the White House said Thursday. Press secretary Josh Earnest told reporters that the numbers to be taken in during fiscal 2016, which starts Oct. 1, represent a “significant scaling up” of U.S. commitments.

The European Union plan, unveiled Wednesday by a top E.U. official, seeks to distribute the share of the unprecedented waves of asylum seekers — many from Syria and Iraq who are seeking havens under the generous social programs in countries such as Sweden and Germany, where Chancellor Angela Merkel was seen taking selfies with migrants.

But the proposal has met stiff resistance in some countries along the migrants’ route from the Mediterranean Sea to Western Europe. This discord reflects widening rifts that have tested European views over humanitarian outreach, cultural diversity and open-border policies.

Similarly, Obama’s plan promptly ran into opposition on Capitol Hill. Michael McCaul (R-Tex.), House Homeland Security Committee chairman, said the Islamic State wants to “sneak operatives into the West” disguised as refugees, and he warned that the United States’ vetting capability is insufficient. “It is heart-wrenching to watch innocent Syrians fleeing the violence in their country, and we can do more to help,” he said in a statement. “But the best way to solve this crisis is at the source.”

Officials in Hungary, the scene of major confrontations between authorities and refugees, said at a news conference Thursday that they plan to make crossing the border illegal and to dispatch the military to enforce the law by the end of the month. They also proposed improving facilities and expanding the number of camps near the border where migrants register for asylum.

On Wednesday, police in Hungary detained a record 3,321 migrants, according to local reports, and more than 42,000 are projected to cross the border over the next 10 days.

About 3,700 more poured into Austria, where the state railway suspended service to and from Hungary because of a “massive overload,” causing some confusion in Budapest, where masses of migrants lined up for hours waiting to board a train.

In Greece, an additional 2,500 migrants were ferried from the island of Lesbos to the mainland for processing. To the north, refu­gee crowds massed along the Macedonia border in a driving rain.

In Denmark, police backed off blockades and allowed refugees to walk through the country to get to Sweden, which has a more-accommodating policy toward refugees.

Polish Prime Minister Ewa Kopacz said her country is willing to accept more refugees, even if that brings additional economic strain.

“Accepting migrants escaping to save their lives is our duty,” she said.

Meanwhile, debate began in the Dutch parliament Thursday about whether the Netherlands should accept the E.U. resettlement plan. Geert Wilders, leader of the far-right opposition, started the session by warning of an “Islamic invasion of Europe, of the Netherlands.”

“Masses of young men in their 20s with beards singing ‘Allahu Akbar’ across Europe,” he said. “It’s an invasion that threatens our prosperity, our security, our culture and identity.”

In a rare acknowledgment of the exodus by the Islamic State, its English-language magazine Dabiq published the widely circulated photo of the body of 3-year-old Aylan Kurdi along with a warning to parents that taking their children to the West was a “dangerous major sin.” The image of the drowned Syrian boy’s body on a beach in Turkey has helped focus international attention on the refu­gees’ plight.

Still, migrants continued to stream into the Keleti train station in Budapest, where their lines were cordoned off as they awaited a trip to uncertain destinations. Police stood outside the train platform, blocking migrants who attempted to board trains traveling to Vienna.

“I don’t know where I’m going. Munich, maybe?” said Majd, 27, who declined to give his last name to protect his family in Syria.

Many had paid smugglers between 200 and 600 euros (about $225 and $675) to transfer them from the Hungarian border to the Keleti station, where volunteers gave out free train tickets. As he waited, Majd said, a man yelled to him, “I hope they catch you!” Meanwhile, a woman from Sweden scoured the crowd and handed out free strollers to mothers carrying small children.

The debates preface the challenges that lie ahead for the E.U. redistribution plan, which requires the assent of all of the countries in the bloc. The proposal also would allow countries to postpone taking in refugees for as long as a year if they had “justified and objective reasons” for doing so, such as a natural disaster, according to a document released after European Commission President Jean-Claude Juncker announced the plan Wednesday. Those nations would pay a fine instead of taking in refugees. It will be discussed among state officials Monday.

In Germany, Merkel called on other European nations to agree to Juncker’s plan.

“If Europe fails on the question of refugees, it will waste a fundamental impulse of a united Europe,” Merkel told lawmakers in a speech in Berlin. Europe needs a “binding agreement,” she said.

Brian Murphy and William Branigin in Washington contributed to this report.

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Today's coverage from Post correspondents around the world