The collision of exasperated migrants with overwhelmed authorities created chaotic scenes Tuesday at choke points up and down the route being traveled by tens of thousands of people seeking refuge in Western Europe.

From the idyllic Greek islands to the fertile plains of southern Hungary, a pileup of people impatient to cross seas and borders produced tense standoffs and desperate flights as migrants sought to bypass registration systems that have broken down amid the crush of new arrivals.

At the Serbian-Hungarian border, hundreds of people chose to dash into a cornfield as police looked on rather than sleep another night on the patch of dirt where they had been confined while they waited to be registered.

Nashat Murad, a 28-year-old lawyer from Damascus, Syria, evaded police by slipping over coils of razor wire at the border, leaving his fingers covered in bright red puncture wounds.

“Just let us cross to Germany,” he said as he jostled with other migrants to board a westbound train at the Budapest station. “We’ve already suffered a lot.”

The refugee crisis spiraled as European leaders prepared to wrangle over a plan that observers say will almost surely fall short.

On Wednesday, European Commission President Jean-Claude Juncker is set to propose a quota system for relocating 120,000 asylum-seekers from the front-line nations of Greece, Italy and Hungary and spreading them across Europe, according to European accounts of the draft plans. Together, Germany, France and Spain would take more than half, according to a draft tally published by Spain’s El Pais newspaper.

The remainder would be distributed across the rest of Europe. But the plan is likely to include an option so that nations opposed to taking in refugees could pay money to help other European Union countries shoulder the burden.

The leaders of the Czech Republic and Slovakia both dismissed calls to accept binding quotas of refugees this week. The Slovaks in particular have said they will take in small numbers and only Christians — effectively rejecting the bulk of the asylum-seekers, who are Muslims from war-torn nations such as Syria.

On the other end of the spectrum, Germany estimates that it may shelter as many as 800,000 asylum-seekers this year. The country’s vice chancellor, Sigmar Gabriel, braced the nation for what could be half a million refugees a year for “several years.”

On Tuesday, German Chancellor Angela Merkel, standing side by side with Swedish Prime Minister Stefan Löfven, called on other E.U. nations to do more.

“We should be clear and to the point,” Merkel said. “I am deeply convinced that this is a task that will decide whether we maintain our European values. The entire world is watching us.”

Reaction to the plan due to be unveiled Wednesday will be a crucial test amid a crisis that has challenged European resolve like few others.

Among the goals of the plan is to create E.U.-run processing centers in front-line nations such as Italy, Greece and Hungary.

Now, the processing of migrants is handled differently from country to country. Many of the systems are rudimentary, with officials using pen and paper to record the presence of thousands of people streaming in from countries across South Asia, the Middle East and Africa.

With tens of thousands of people traveling the well-worn path northward from Turkey into Europe, the process has broken down, leading to growing frustration among migrants worried that delays will jeopardize their chances of making it to Western Europe.

“What you have are bottlenecks caused by the need to register people. It’s part of the legal process, and the police are just doing their job,” said William Spindler, a spokesman for the U.N. High Commissioner for Refugees (UNHCR). “But the problem is that they don’t have the capacity to register more than a few hundred a day.”

Perhaps no place in Europe has felt the consequences more acutely than the Greek island of Lesbos, where the UNHCR said Tuesday there were 20,000 migrants jammed into woefully inadequate camps.

In recent days, their frustration has boiled over into demonstrations, with some migrants setting fire to trees and trash as they demanded to be allowed to board ferries to Athens. Police have used batons to beat them back. Greece’s migration minister, Yannis Mouzalas, told a radio station Monday that the island was “on the verge of exploding.”

Overnight, the government rushed emergency aid to Lesbos and streamlined the registration process. Elisabetta Faga, field coordinator for Doctors Without Borders, said the result was a calmer situation Tuesday, though she stressed that authorities still needed to find a more permanent solution.

“These are temporary, emergency measures,” she said. “The existing physical and human infrastructure for dealing with this number of arrivals is inadequate.”

She said the facilities for sheltering migrants are “about a 10th of the size of what they need to be.”

Further along on the migrant trail, there was also a massive backup of migrants at the Greek-Macedonian border. On Monday, a record 7,000 people crossed — including 4,000 who came streaming into Macedonia in just one chaotic hour, according to Alexandra Krause, a UNHCR representative at the crossing.

On Tuesday, she said that 4,800 people had crossed by mid-afternoon but that the situation was calmer because people were coming over in small groups. Still, she said the mood remained tense and migrants were impatient to be on their way because they fear that Hungary may soon block their path.

“There are a lot of rumors floating around,” Krause said. “People are anxious that the border might be closed.”

Hungary’s nationalist prime minister, Viktor Orban, did nothing to tamp down those fears Tuesday. After a quick visit to the border Monday night, he told a Hungarian newspaper that the government would speed the construction of a fence along the country’s border with Serbia. Orban has also said the military may soon be deployed.

For now, thousands of people continue to stream across each day, a flow that has overwhelmed the police force’s ability to cope.

A new registration center was packed to capacity within hours of its opening Sunday, and thousands more migrants had to sleep in an informal, open-air camp by the side of the road Sunday and Monday nights.

With overnight temperatures falling into the 40s, some of the migrants revolted, pushing through police lines Monday and Tuesday in an effort to move on toward Budapest without first registering.

Hassam Badawi was among them. After fleeing the police, he paid a smuggler 300 euros ($335) to drive him to the capital. By Tuesday afternoon, he was boarding a westbound train — and counting down the minutes until he could leave Hungary behind.

“It was cold. It was dirty. I couldn’t sleep,” said Badawi, a 43-year-old lawyer from Damascus. “Even crossing the sea was better than the Hungary border.”

Faiola reported from Berlin. Michael Birnbaum in Brussels and Gergo Saling in Budapest contributed to this report.

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