BRUSSELS — Even as three more countries followed Germany in introducing border checks to control a flood of migrants, the European Union on Monday failed to agree on a modest plan that would force individual countries to take in a share of some of the hundreds of thousands now seeking asylum in Europe.

Gathering in Brussels for an emergency meeting, interior ministers from across Europe agreed to share 40,000 migrants sheltering in Greece and Italy, but only on a voluntary basis, a watered-down version of a plan announced in May.

But as the fractious meeting stretched into the evening, there seemed little prospect that ministers would endorse a new plan put forward last week by Jean-Claude Juncker, president of the European Commission, for a program of resettlement for a further 120,000 asylum seekers that would be compulsory for member countries.

Jean Asselborn, the minister of foreign affairs of Luxembourg, which holds the union’s rotating presidency, told a news conference late Monday that a majority of countries accepted “in principle” Mr. Juncker’s plan, but added that the discussions had been “very difficult.”

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Discussions will not resume until next month, a blow to Mr. Juncker, who last week pleaded for “immediate action.”

In a sign of the disharmony caused by Europe’s worst humanitarian crisis since World War II, ministers did not issue a joint final statement as is customary and left Luxembourg to issue a summary of the discussions in its own name.

The rancorous haggling in Brussels over the distribution of 160,000 migrants — a small part of the total — played out as Austria, Slovakia and the Netherlands introduced border controls on Monday, after a decision by Germany on Sunday to set up checks on its own southwestern frontier and halt train traffic with Austria.

The reintroduction of border controls, described as a temporary measure to restore order to an often chaotic flow of migrants, was the most serious challenge in years to Europe’s cherished system of passport-free travel across much of the Continent.

The 26 European countries that are party to the so-called Schengen Agreement, a cornerstone of European integration that enshrines open borders but also allows for temporary controls for security reasons, have in the past periodically reinstated checks but never because of pressures from migration.

The reintroduction of controls also threatened to create an unpredictable patchwork of complications and potentially risky obstacles for migrants seeking to make their way through Europe to preferred destinations in places like Germany or Sweden, where benefits are greater and the processing of asylum applications moves faster.

The desperation to reach such countries, even as border controls have been tightened, has increasingly driven migrants and refugees into the hands of smugglers, leading recently to the deaths of 71 migrants who suffocated in the back of a truck along a highway in Austria.

In Hungary, the authorities said that a near-record 5,353 migrants had crossed into the country from Serbia before noon on Monday — even as Budapest continued to try to seal off that border, which is being reinforced with the construction of a 109-mile fence topped with razor wire.

About 50 police officers, wearing riot gear and equipped with gas canisters, converged Monday on the train tracks linking the villages of Roszke, Hungary, and Horgos, Serbia, which thousands of migrants had used to cross the border in recent days. An official in a bright yellow jacket turned away migrants seeking to enter Hungary.

The border measures raised pressure on participants at the emergency meeting in Brussels to close the deep fissures that have opened up among European nations over how to handle the migrant crisis. A further note of urgency came with an appeal in Brussels made by the aunt of Aylan Kurdi, the 3-year-old Syrian boy whose body washed up on a beach in Turkey this month. Photographs of the dead child shocked European public opinion and helped set off an outpouring of sympathy for migrants.

“Europe has not done enough,” the aunt, Tima Kurdi, a resident of Canada, said during a visit on Monday to Brussels for talks with officials. “Germany took the biggest number, and now it has too many. Every country has to take responsibility. Aylan’s death was, I believe, a message from God to the world to wake up and do something about these refugees. Everybody is closing the door in their face.”

All the same, ministers from several East and Central European countries remained steadfast in their opposition to the compulsory distribution of migrants proposed last week by Mr. Juncker.

“This proposal is not solving the problem,” said Robert Kalinak, the Slovak interior minister. The Czech Republic, Hungary, Romania and the Baltic States have all voiced deep reservations about taking in refugees, arguing that they have no tradition of offering refuge to people of different cultures; that their economies cannot sustain the influx; and that most of the migrants want to live in richer and more welcoming places like Germany and Scandinavia.

In a dig at former Communist countries in the East that have said they will take only Christian refugees, Mr. Asselborn, the Luxembourg minister who ran Monday’s meeting, said that even his own country, the European Union’s smallest, “could take in a few hundred people who are not necessarily Christians, who have a different skin color, and that should be able to work in big countries like Poland or the Czech Republic or in Slovakia.”

In an effort to win over opponents of the plan, Germany, France and other nations embraced some of the Eastern Europeans’ tough language, calling for tighter controls of Europe’s external borders and firmer measures against migrants who fail to qualify as refugees, including their swift return to countries deemed safe.

The French interior minister, Bernard Cazeneuve, stressed that economic migrants needed to be separated from genuine refugees fleeing war or oppression. “There cannot be humanity without firmness and responsibility,” he told reporters.

Ministers endorsed plans to increase cooperation against smuggling and border violations with Turkey, which has become one of the major transit countries for migrants making their way to Europe.

In a speech on Wednesday to the European Parliament, Mr. Juncker presented the migrant crisis as a test of Europe’s ability to take common action and asked that interior ministers move swiftly to endorse his relocation plan at their Monday meeting. Voicing doubts that a voluntary program would work, he said, “This has to be done in a compulsory way.”

The European Commission did not include the remark in an official transcript of the speech, in what could be a sign of the strength of opposition to this initiative.

Luis Morago, campaign director for Avaaz, a refugee advocacy group, said that Europe’s reluctance to adopt mandatory quotas did not necessarily emasculate the relocation effort, but that “in practice the only way of getting many European governments to do anything is if it is compulsory.” Europe’s halting response to the migrant crisis, he said, had so far been “disappointing and contradictory.”

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Elizabeth Collett, director of the Migration Policy Institute Europe, a research group in Brussels, said Europe’s difficulties in finding a common approach reflected the flaws in a shaky European system whose rules and institutions were “only half built” and whose ultimate goal — a fully integrated European state or simply a collaborative jumble of distinct nation states — “has never been decided.”

On migration, she added, “there is simply no consensus.”

“Some countries accept the idea that Europe is a place for asylum seekers,” she added, “but others don’t think this is their responsibility.”

The European Union’s mixed signals have exasperated nations that share a border with the bloc and that have become transit routes for ever growing numbers of Syrians, Afghans and others seeking entry into Europe.

Citing Hungary’s decision to make unauthorized entry into the country a criminal offense starting Tuesday, Serbia said it would set up reception centers in the north of the country and pleaded for the European Union, of which it is not a member, to take action.

While Berlin said the controls along the German-Austrian border were only a temporary emergency measure, the restrictions, a response to the strain on local communities, signaled that Chancellor Angela Merkel’s welcoming stance toward the migrants was encountering domestic resistance.

“There is no order, there is no system, and in a country governed by the rule of law, that is a cause for concern,” Horst Seehofer, the governor of Bavaria, a deeply conservative state in the south, said Sunday.

Joachim Herrmann, the Bavarian interior minister, told a local television station, “We need better controls in general because we have determined that in recent days many of those on the move are really not refugees.”

Officials in Eastern and Central Europe, including Hungary, have made similar arguments as they battled to scuttle Mr. Juncker’s plans for the swift distribution of 160,000 migrants.

“A mandatory quota for the E.U.-wide relocation of migrants is unlikely to be achieved quickly, if at all,” said Carsten Nickel, a senior vice president at Teneo Intelligence, a political risk consultancy.

Dimitris Avramopoulos, the European Union commissioner for migration, insisted that action to relocate migrants was still possible.

“The majority of member states are ready to move forward — but not all,” Mr. Avramopoulos told a news conference. “The world is watching us. It is time for each and every one to take their responsibilities,” he added.