Thousands of migrants poured into Austria on Sunday, deepening a rift among European Union members as neighboring countries sought to push more of the burden of the human wave onto one another.

As some 21,000 people crossed into Austria from the southeast over the weekend, governments in the region quarreled publicly over how to respond to the crisis. The government in Vienna warned that it was inundated and would send migrants back to Slovenia and Croatia—where it said people should be required to seek asylum instead of being passed on to more affluent Western European countries.

“This is no longer a pursuit of safety, rather a way to optimize asylum,” Austrian Interior Minister Johanna Mikl-Leitner said

She urged European countries, “even the less attractive” to arriving migrants, to share the burden and take in more people. “Then it will be demonstrated clearly: In Europe, there is the right to protection but not the right to pick the most attractive country for it,” the minister said.

The crisis has intensified in recent weeks, as the tide of refugees and other migrants seeking a better and safer life in Europe grows and many are avoiding requesting asylum in EU countries along their way to Germany and other wealthier European countries. Overwhelmed countries in Central and Southeastern Europe have made efforts to speed up the migrants’ travel to the west and north—or to detour them to neighboring countries—prompting the squabbling among governments.

About 10,000 migrants had arrived in the Austrian border town of Nickelsdorf by late Sunday after approximately 11,000 entered the country on Saturday, a spokesman for the Austrian Interior Ministry said.

To the south, the desperate wave of migration continued. Croatia said Sunday about 3,900 migrants were making their way across the border from Serbia at the crossing in the town of Tovarnik.

As a result, Croatian officials said they would send the newest arrivals by bus and train to border crossings with Hungary, which in turn called the plan “cynical and irresponsible.”

“Human smuggling is done in Croatia with the help of the government,” said Zoltan Kovacs, a spokesman for the Hungarian government.

Croatian Foreign Minister Vesna Pusić said in an interview Sunday that Hungarian officials “got a bit overexcited about the situation,” but that “on the ground, they are helping.” She said Croatia could cope with thousands, but not with tens of thousands of migrants, who all want to go to Germany anyway.

“We got 27,000 people in five days. Ms. Pusić said. “We are treating then with respect but there is no way we can register them all.”

Many countries, particularly in the EU’s east, continue to oppose a plan made in Brussels that would resettle migrants proportionally based on each country’s population and economic power. Hungarian Prime Minister Viktor Orban and Poland’s conservative opposition leader, Jaroslaw Kaczynski, recently put the blame for the recent surge in arriving migrants on Germany, which last month declared it would take in refugees from Syria, effectively suspending the application of Europe’s strict asylum rules.

They said that instead, Europe should strengthen its external borders to discourage migration. Hungary last week decided to extend its border fence with non-EU member Serbia to parts of its border with fellow EU members Croatia and Romania, which are outside Europe’s open-border Schengen area.

Tensions between Hungary, Croatia and Romania rose amid the crisis, with Hungary declaring over the weekend it could prevent Croatia from joining the Schengen area, while Romania sharply criticized Hungary for erecting the fence. Romanian Foreign Minister Bogdan Aurescu said the barrier was “an autistic solution, which is unacceptable and not in line with the European spirit.”

Hungarian Foreign Minister Peter Szijjarto retaliated on Sunday.

“We would expect more modesty from a foreign minister whose prime minister is currently facing prosecution,” Mr. Szijjarto said in a release from his ministry Sunday. He alluded to corruption charges pressed against Romania’s Prime Minister Victor Ponta, who was indicted earlier this year and denies the allegations.

“We are a more than 1,000-year old state, which had to defend several times not only itself but also Europe over the course of its history. This will be so this time around too, even if the Romanian foreign minister likes it or not,” Mr. Szijjarto added.

Budapest has summoned a number of ambassadors from countries that have made critical comments about the way it has handled the refugee crisis.

Meanwhile, it reopened the border crossing at Roszke with Serbia on Sunday on the condition that the country ensures no migrants enter Hungary at that crossing.

Romania on Sunday said it was holding an exercise in preparation for a possible influx of refugees. Dozens of army tents and portable toilets were being set up since Saturday evening at Lunga, on the Serbian border, with gendarmes and emergency response teams deployed.

EU leaders will attend a summit on Wednesday to discuss how to deal with the influx of migrants. Member states have yet to agree on how to distribute 120,000 people seeking asylum in Greece, Italy and Hungary more evenly across the bloc. Hungary, however, said it doesn’t want to be part of the plan, so its share could now be divided between Italy and Greece or between Croatia and Slovenia, an EU official familiar with the talks said. Wednesday’s summit will follow talks among EU interior and migration ministers planned for Tuesday.

Another EU official said the European Commission on Wednesday was set to launch a series of inquiries into breach of EU asylum law in several EU countries, including Hungary. The so-called infringements take years to settle and can end up in court and with fines.

Speaking on Saturday, German Economics Minister Sigmar Gabriel said the international community needed to invest a minimum of €5 billion ($5.64 billion) to alleviate the refugee crisis in the Middle East, the source of the migration to Europe.