— Pressure is mounting on Germany and other nations to scale back their generous policies welcoming refugees, with opponents, including some of the region’s most influential leaders, arguing that the promise of aid is enticing more and more asylum seekers to make a break for ­Western Europe.

There was no sign Monday that the crisis was easing, with Britain and France — two nations criticized for not doing enough — pledging to take in tens of thousands of asylum seekers. But Germany — the nation taking in the lion’s share, an estimated 800,000 by year’s end — continued to lead the way, pledging to hire 3,000 more police officers and spend $6.7 billion more to address the crisis, including emergency housing for 150,000 people.

Yet even as German Chancellor Angela Merkel said her nation should be “proud” of its response, other European leaders and domestic critics blamed Germany, as well as similarly generous nations such as Sweden, for offering benefits so lucrative that they had become an incentive for asylum seekers to risk their lives over land and sea.

The sniping came as the pace of arrivals accelerated Monday. In Greece, the refugees’ first port of call, authorities requested emergency European Union assistance as islands received asylum seekers faster than they could be ferried to the mainland. Greece’s coast guard said it had rescued more than 2,000 asylum seekers in the Mediterranean Sea since Friday. And hundreds of migrants scuffled with Hungarian authorities on the border with Serbia before pushing into the country on foot.

Germany responded to the criticism Monday by announcing a reduction in cash handouts for asylum seekers during their initial months of processing, instead saying it would offer them more food stamps and in-kind aid. Berlin also said it would push to have western Balkan countries such as Kosovo declared “safe” in a bid to weed out the many thousands of migrants now claiming asylum from countries not at war.

The German maneuvers reflected the complex nature of Europe’s migrant crisis, in which desperate Syrians and Iraqis are searching for sanctuary in the wealthy countries of Europe’s core along with a host of economic migrants pouring in from countries as far-flung as Pakistan and Bangladesh.

“We want to reduce the number of pull factors, and I think it’s a big step forward that we have consensus in our government to reduce the monetary benefits for those seeking asylum,” said Stephan Mayer, a German national lawmaker and home affairs spokesman for the Christian Social Union, part of Merkel’s ruling coalition. Referring to criticisms by European leaders including Britain’s David Cameron and Hungary’s Viktor Orban, he said, “I can’t say that Orban or Cameron are completely wrong.”

In the crowded refugee centers across this nation of 81 million, asylum seekers have conceded that they have come to Germany because it is doing more to help than other nations in the region.

Mohammed Mazher Alkilany, 28, a former PR consultant for the Damascus tourism board who is living in a temporary shelter in east Berlin, said his family of three is living on 233 euros a month provided by the government — a sum he described as too little to cover the cost of warm clothes and blankets for the coming winter.

But they are also living in free temporary housing in a building outfitted with a playground and rooms with shared kitchens, bathrooms and washing machines. He insisted, though, that he did not come to Germany simply for its generous benefits.

“I came here because Germany is safe; there is no war,” he said. “Germany is the best in Europe. France is no good, you cannot get language classes there, but in Germany you can learn the language for free.” Although Sweden is offering similar aid, he said it was “too far away, it is very cold, and it is always night there.”

A few European nations have been willing to set up operations to legally and safely bring, for example, Syrian refugees directly from bordering nations such as Turkey and Lebanon. But they have put strict limits on numbers, with all 28 E.U. nations offering just over 53,000 such spots since 2013, according to U.N. figures. That is a drop in the ocean compared with the more than 4 million Syrian refugees.

Instead, European nations have preferred to deal with ­asylum seekers only at the point when they are politically forced to — after the refugees physically cross their borders. Analysts say that a shift in policy would reduce the incentives to make the dangerous voyage to Europe’s shores but that there was little political support for such an effort.

Cameron announced Monday that Britain would resettle 20,000 Syrian refugees directly from the Middle East over the next five years — a figure equal to the number of asylum seekers Germany took in over the weekend.

Cameron, however, said Britain was nevertheless acting with “head and heart” by accepting refugees only from camps around the Syrian border, while seemingly taking a jab at nations such as Germany for encouraging illegal trips by accepting so many.

“We want to encourage people not to make that dangerous crossing in the first place,” Cameron said.

Up to now, Britain has resettled only 216 Syrian refugees through its government program.

The scant opportunities to obtain visas on the ground near the Syrian conflict were dramatized by the drowning death last week of 3-year-old Aylan Kurdi, who had family in Canada but whose parents had been unable to get family reunification visas that would have given them a legal route out of Turkey. Instead, they tried to reach Greece by boat, with tragic consequences.

“We would prefer that no refugee would have to take that dangerous journey to have to reach safety in Europe,” said Melissa Fleming, a spokeswoman for the U.N. High Commissioner for Refugees.

Under a European Commission proposal to be released Wednesday, reception camps would be set up in Italy, Greece and Hungary to route new arrivals to other European countries. That could eliminate most of the risky overland legs of the journey. But the incentives to set sail from Turkey, Libya or Egypt would remain.

French President François Hollande said Monday that his nation stood ready to take in 24,000 refugees as part of the plan. Spain’s El Pais newspaper published leaked numbers that show Germany taking more than 31,000, while Spain would take in nearly 15,000. Other countries would take in far fewer.

The debate over whether Germany is being too generous came as more European politicians are questioning the open-door approach. “Just between us, you know, the problem is not a European problem. The problem is a German problem,” Orban said last week after negotiations with E.U. leaders. In France, which has struggled with how to respond to the influx, far-right leader Marine Le Pen took a pot shot at Berlin’s efforts.

“Germany is probably thinking of its moribund demographics, and it is probably trying to lower salaries again and to continue to recruit slaves via massive immigration,” she told a meeting of her National Front party.

The move in Germany to reduce cash handouts for asylum seekers came as the country is also seeking to rapidly expel those who do not qualify. During the first six months of this year, for instance, about 45 percent of people seeking asylum in Germany were Europeans from the Balkans.

Overall, however, Germany is dramatically ramping up its spending on the crisis, with the nation preparing for years, if not generations, of societal change from the arrival of so many newcomers.

“What we are experiencing now is something that will continue to preoccupy and change our country in the coming years,” Merkel said Monday. “And we want it to change in a positive way. And we think we can make that happen.”

Birnbaum reported from Brussels. Stephanie Kirchner in Berlin and Karla Adam in London contributed to this report.

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