LONDON — A relentless stream of migrants continued to flow into Europe on Monday, as France, Germany and Britain moved to offer new assistance to try to deal with an influx that is severely testing the Continent’s ability to respond.

President François Hollande of France announced on Monday that his country would take in 24,000 asylum seekers over two years, Britain said it would take in 20,000 refugees from Syria, and Chancellor Angela Merkel said Germany would set aside 6 billion euros, about $6.7 billion, to deal with the crisis.

Demonstrating the extent to which countries across Europe are strained by the migrant influx, there were signs of tension across the Continent.

In the southern Hungarian town of Roszke, near the border with Serbia, hundreds of refugees who had managed to cross into Hungary were herded into a trash-strewn field to wait, some for two days, to be transported to so-called reception camps.

Several hundred migrants formed into a marching group and tried to push their way out of the makeshift camp, chanting “Freedom! Freedom!” before they eventually retreated when the police warned they could be arrested.

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“It’s a big mess, as you can see,” said Balazs Szalai, a volunteer with a refugee assistance group.

In Spain, the police fired rubber bullets to regain control of a detention center in the eastern city of Valencia, where about 50 migrants tried to escape, The Associated Press said, citing local news reports. The episode began late Sunday, when migrants assaulted a guard and grabbed his keys, according to The A.P.

And in Greece, the authorities said they had requested European Union aid to help it cope with the surge in the number of migrants arriving each day, often on rickety boats from Turkey.

Mr. Hollande said a plan that was expected to be presented on Wednesday by the European Commission, the union’s executive arm, would redistribute 120,000 people across the bloc over the next two years.

“It is the duty of France, where the right to asylum is entirely part of its soul, of its flesh,” said Mr. Hollande, who added that France was ready to host an international conference on the matter.

The United Nations refugee agency has said that more than 310,000 people have crossed the Mediterranean to reach Europe since the beginning of the year, prompting Mr. Hollande to say that the situation was “dramatic” and “serious,” but that it could and would be brought under control.

The seemingly disparate projections of the numbers of migrants cited by different governments underlined the confusion and difficulty that the European Union has found in trying to come up with a coherent policy for the fast-moving situation.

The European Union needs to provide “massive humanitarian aid” to countries like Jordan and Lebanon that have taken in millions of Syrian refugees, Mr. Hollande said.

The French leader added that the European Union needed to create so-called hot spot reception centers in countries like Greece, Hungary and Italy to identify and register migrants as they arrived in the European Union and to turn back those who do not fulfill the requirements for asylum.

It was not clear how such centers would operate.

In Britain, Prime Minister David Cameron detailed his government’s plan to accept thousands more refugees from Syria, after a public and political uproar over his initial reaction to the crisis.

Mr. Cameron said on Monday that he would accept up to 20,000 Syrian refugees in Britain, but that they would most likely be limited to asylum seekers from camps in Jordan, Lebanon and Turkey, rather than in Europe.

The British government has indicated that it does not want to add incentives, or “pull factors,” that would encourage more migrants to risk the passage to Europe, nor to favor those migrants who could afford to pay people smugglers over those who are in the regional camps.

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With people inside and outside Mr. Cameron’s governing Conservative Party critical of the European Union, Britain will most likely continue to reject the idea of mandatory quotas to distribute migrants and asylum seekers already in Europe across member states.

Mr. Cameron announced last week that Britain would add an additional 100 million pounds, about $150 million, to the £900 million it already provides for humanitarian aid to displaced Syrians.

Britain will also take some of the funds that it usually sends abroad and use that money to house and help Syrians in their first year in Britain, the chancellor of the Exchequer, George Osborne, said on Sunday. He refused to confirm a specific figure that would be spent on the migrants.

In Austria, migrants continued to arrive, despite an announcement by Chancellor Werner Faymann on Sunday that the country planned to limit the number it would allow in.

The police along Austria’s eastern border with Hungary, where thousands of migrants were stranded last week before being allowed to cross, were increasing checks in the area in an effort to find human traffickers, but they allowed migrants entry, Reuters reported.

Germany, which is expected to receive 800,000 migrants this year, announced its €6 billion financial commitment along with other plans to absorb the huge influx.

According to official German figures, about 40 percent of those who have applied for asylum are from the western Balkans and are unlikely to have their applications accepted. About 49 percent of the migrants coming to Europe by sea this year have been from Syria, the United Nations relief agency said, but there are no reliable figures for those coming by land.

“What we are experiencing now is something that will occupy and change our country in coming years,” Ms. Merkel said on Monday after a meeting of government leaders on possible measures to cope with the influx. “We want that the change is positive, and we believe we can accomplish that.”

In addition to the increased funding, the German government plans to pass several laws in the coming month aimed at speeding the processing of applications and at bolstering efforts to get newcomers into jobs or schools.

Germany and France are pressing for a quota system by which European Union countries would accept migrants according to their populations and relative wealth.

The quota system has been steadfastly rejected by many countries, which argue that immigration is a matter of national policy and sovereignty and should not be determined by Brussels.

On Monday, the Hungarian prime minister, Viktor Orban, criticized the bloc’s proposals for quotas, saying that Hungary was a “black sheep” and would not follow the “flock” of other European Union countries, The A.P. reported.

Mr. Orban told a group of Hungarian diplomats that the proposed redistribution system made no sense given the bloc’s border-free system, which, he said, would make a quota system impossible to enforce.

“How is this going to work?” he asked. “Has anyone thought this through?”

Mr. Orban also repeated a warning he has made to the thousands of migrants on their way to Europe. “Don’t come!” he said in a speech to his ambassadors gathered in Budapest. “I ask those who want to go to Europe through Hungary not to come.”

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