At times, the migrants -- most of them from Iraq, Syria and Afghanistan -- tussled with police blocking a road from this holding site to a transit camp near Roszke, Hungary, where they can register as refugees and continue their journeys.
Buses were carrying small numbers of migrants to the camp, but many have been forced to wait at the holding site for as many as three days with little in the way of services or support.
One Hungarian nonprofit was on site handing out biscuits, fruit and water, and a medical tent was erected Monday.
Meanwhile, Austria and Germany warned they can't keep up with the influx of refugees and said they must begin to slow the pace.
More than 16,000 migrants have streamed into Austria since Saturday, Burgenland state police spokesman Wolfgang Bachkoenig said Monday. Virtually all continued to Germany, where the city of Munich had received more than 17,500 people, police said.
"We must now, step by step, go from emergency measures to a normality that is humane and complies with the law," Austrian Chancellor Werner Faymann said.
The United Nations' refugee agency, UNHCR,
estimates that more than 366,000 refugees and migrants have crossed the Mediterranean Sea to Europe this year.
At least 2,800 have died or disappeared during the journey. Those who make the crossing face uncertain futures in European nations, which differ in their approach to asylum seekers.
Some European nations offer help; Denmark tightens restrictions
On Monday, German Chancellor Angela Merkel said her country would pledge an additional €3 billion to the migrant crisis. She said Germany "is of course willing to accept more refugees," but called on other European countries to take in more.
Moments after Merkel spoke, French President Francois Hollande said France is ready to take on more responsibility.
He said the European Commission will propose distributing 120,000 refugees over the next two years, of whom France would take in 24,000.
"We will do so because it is the principle to which France is committed," Hollande said.
And Britain will take up to 20,000 Syrian refugees over the next five years, Prime Minister David Cameron said Monday. But the country will focus on resettling refugees from camps in countries bordering Syria, not those who have already entered Europe.
"This provides refugees with a more direct and safe route to the United Kingdom rather than risking the hazardous journey to Europe, which has tragically cost so many lives."
Refugees will receive a five-year humanitarian protection visa, he said.
Not every European country is opening its arms.
Denmark, for instance, paid for ads in Arabic in four Lebanese newspapers to get the word out about its own new, tightened restrictions -- such as reducing social benefits -- to try to prevent refugees from getting into the Scandinavian nation.
"We cannot simply keep up with the present flow," Immigration and Integration Minister Inger Stojberg, a member of the right-wing Venstre Party, said on Facebook. "In light of the huge influx to Europe these days, there is good reason for us to tighten rules and get that effectively communicated."
'We went through a torture'
Many of the migrants arrive with harrowing tales of crossing the Mediterranean, then walking from Greece through Macedonia, Serbia, Hungary and finally into Austria.
Austria's border with Hungary remains open to potential refugees, Austrian Interior Ministry spokesman Alexander Marakovits said Sunday, as packed buses and trains continued to arrive.
Many Austrians brought food and water and cheered for the refugees pouring onto the platform at Vienna's train station.
One man who recently arrived in Austria talked of the family's difficult journey through Hungary.
"We went through a torture," he said, standing next to his two daughters. "We walked 110 kilometers (70 miles) with the children. They didn't allow us to take cars or trains."
But he said the Hungarian people "were very nice" and the situation got better when the family arrived in Austria.
"We are comfortable here, and we like the people and the government of Austria."
Of the thousands who arrived in Austria this weekend, a dozen or so have opted to apply for asylum there, the country's Interior Ministry said. Many want to go farther, particularly to Germany.
Attacks against refugees on the rise
But the dangers don't stop once refugees reach Germany.
At least 340 attacks have taken place on refugee camps in Germany this year, the Interior Ministry said. Most of the incidents are believed to be fueled by radical right-wing, anti-immigrant sentiment.
The attacks include vandalism, hate speech and arson, as well as violent attacks on people. At least 38 violent assaults have been recorded this year, up from 28 last year.
On Monday, another suspicious fire broke out at a house for asylum seekers in Rottenburg am Neckar, police said.
Five people were injured -- three by smoke inhalation and two by jumping out of the building's first and second floors. None of the rooms is inhabitable anymore.
The cause is under investigation.
European Union countries have an open-border policy that allows the free movement of people between member states. While Germany, France and other countries are opening their doors to more migrants, countries such as Hungary and Austria are clamping down on the flow.
Hungary's right-wing government, trying to stop the flood of migrants, has erected a barbed wire fence along its more than 160-kilometer (100-mile) border with Serbia to prevent them from crossing there. Serbia is not an EU country.
In Austria, the Interior Ministry warned that it is illegal to drive across the border to Hungary, pick up a group of migrants and transport them back to Austria.
But some people are doing exactly that.
A group of volunteers from Austria, Germany and Slovakia -- organized on Facebook -- formed a convoy of almost 200 cars to shuttle migrants from the Hungarian border.
"We think that around 380 people came with us," volunteer Erzsebet Szabo said Monday morning. "We are very happy."
Szabo said she's not afraid of getting arrested. After all, she said, even that fate wouldn't compare to what the refugees have endured.
"It's very important that we, altogether, give this big sign that refugees -- the people that need our help and come from the war -- have our solidarity and support."
The disparate responses have resulted in calls from United Nations and European Union officials for European countries to stand shoulder to shoulder in their response to the crisis, the like of which has not been seen since World War II.
"We need concrete, coherent, rational political decisions in the sense of solidarity and responsibility, because European Union was built in decades after the second world war on the experience of that war, that made many of our Europeans flee and leave Europe," EU foreign affairs chief Federica Mogherini told CNN's Christiane Amanpour in an exclusive interview Monday.
"Now we should remember our story and act following the same values and principles that have allowed us to build a continent in peace and prosperity," she said. "We are rich. We are in peace. We have the duty to save and protect people that are fleeing from war."