Migrants streamed into Budapest's Keleti train station on Thursday, breaking a deadlock with authorities outside. Trains are running, but the borders with Austria and Slovakia are closed, a rail official said, limiting where the refugees can travel.

[Previous story, posted at 6:07 p.m.]

[CNN] -- A tiny step in dealing with Europe's deadly and massive migrant crisis was attempted Wednesday as Italy, Germany and France sought a united response to the worsening plight of millions of refugees.

Foreign ministers Paolo Gentiloni of Italy, Frank-Walter Steinmeier of Germany and Laurent Fabius of France presented the European Union with a joint document calling for a revision of asylum rules and a fairer distribution of refugees, according to the Italian Foreign Ministry.

The effort comes as the photo of a toddler's lifeless body -- one of 12 people who drowned off Turkey and washed up on a beach -- has come to symbolize the gravity of the crisis.

The European Union foreign policy chief, Federica Mogherini, urged "united" action and said the issue will be part of the next EU foreign ministers meeting on Friday and Saturday. The EU Commission was meeting to prepare new proposals.

Another batch of refugees, meanwhile, arrives in Munich with almost every train that pulls into the station.

Tired, hungry, weak from the heat, they speak of a harrowing journey -- and the joy of finally making it to Germany.

But in Budapest, the scene is more desperate.

Refugees in the Hungarian capital held up scraps of paper: "Help Syrians," they read. "Babies are tired."

Still, almost anything -- even this -- is better than the chaos and killing from which they have fled: in Syria, where cities lie in rubble; in South Sudan, ravaged by war and poverty; in Libya, where warlords maraud and people suffer; in Iraq, where ISIS likes to videotape its slaughter of innocents.

These lands are no place for families to thrive. No place to raise one's children.

In Budapest, the migrants -- free at last of bombing and war -- lie listlessly, waiting to board trains for Western Europe.

"We hope you will save us," a Syrian refugee named Houriye told CNN. "I beg you to save us."

But the migrants are denied by authorities. Hundreds wait. Still, they have experienced much worse than this.

Mahmoud, a chemical engineer, said he was a successful businessman until he lost it all to Syria's war. The face of his 4-year-old son is scratched from a fall at a border crossing.

"It's too tough for me to see my family like this," he said.

Wildly varied response

The response of various European governments has varied wildly.

JUST WATCHED Flood of migrants arrive in Germany after grueling trek Replay More Videos ... MUST WATCH Flood of migrants arrive in Germany after grueling trek 01:59

On Thursday, Hungarian Prime Minister Viktor Orban plans to meet with other members of the European Union to figure out how to cope with the emergency. His nation is a transit point for migrants trying to make their way north. Hungary has responded by erecting a fence along its border with Serbia.

In Germany, the interior minister will address Parliament, after a planned asylum center was burned down.

Germany's government has been more accepting of asylum seekers, but political leaders there have had to contend with xenophobic protests.

On Tuesday, police arrested a suspect in an attack earlier in the week on about 40 asylum seekers in a shelter in Brandenburg.

And in France, migrants shut down the Eurostar cross-channel service between England and France overnight as they poured onto the tracks near Calais, the French entry to the channel tunnel. Trains between London and Paris and London and Brussels were turned back.

Caught in the middle are the desperate men and women, with children in tow, who have fled wars in Syria and Iraq in overcrowded, sometimes deadly voyages by land or by sea.

In just the latest example, at least 12 people traveling on boats filled with refugees drowned off the coast of Turkey, Turkish officials said, according to the country's semiofficial news agency, Anadolu.

The incident took place after two boats left Bodrum and were trying to reach the Greek coast, Anadolu reported.

Passports and visas, please

At Keleti station in Budapest, tensions simmered

Hundreds of people who had expected to board trains to Austria and Germany found Hungarian police officers barring their way.

They want to make their way to Western Europe, where they hope to claim asylum.

The journey for almost all of them has been arduous.

"We have been here five days. No food, no sleep -- no place to sleep, no anything," one Syrian refugee at Keleti station told CNN on Tuesday.

JUST WATCHED Hundreds of refugees arrive in Germany from Hungary Replay More Videos ... MUST WATCH Hundreds of refugees arrive in Germany from Hungary 02:14

He and other travelers had train tickets but were not being allowed through, he said.

"The problem is the amount of migrants with the wrong papers," said Reka Hegedus, a spokeswoman for the train station. "Obviously, even if they have tickets, it is not enough."

Only those with proper documentation -- a valid passport, a ticket and any necessary visas -- were being allowed into the station, with police checking the papers of those seeking to enter.

"European regulations require that a person wishing to go on to Austria or Germany, for example, has the necessary documents," a government statement said. "People at Budapest's Keleti railway station demanding to be allowed free passage are demanding something which is not possible under European legislation."

A Hungarian government representative, János Lázár, requested that the German embassy provide information to migrants at Keleti rail station.

Italian authorities have temporarily reinstated border controls at the Italian-Austrian line in the region of Alto Adige in northern Italy, after being requested to do so by Bavarian authorities in Germany who say they are "overwhelmed" by the influx of migrants, according to a statement released by Italy's Bolzano prefecture.

Bavaria has had a great number of refugees arriving mainly from the Balkan route and the situation is getting difficult to handle, the Italian statement said.

"Work is in process to find new shelters for the continuously growing number of refugees," the statement said.

The governor of Italy's South Tyrol, Arno Kompatscher, has asked the Italian government for permission to help Bavaria, as part of "a strategy of European openness that allows to travel beyond regional and national limits," the statement said.

The measures mean that those wanting to cross the border will need to provide the appropriate paperwork or identification. In the meantime, the region of Alto Adige will host 300 to 400 migrants or refugees with shelter and food.

Flashpoint: Hungary

Hungary has come under increasing pressure from Western European countries to change its approach to the migrant crisis.

JUST WATCHED Hungary responds to criticism of razor-wire fences Replay More Videos ... MUST WATCH Hungary responds to criticism of razor-wire fences 03:33

It has been erecting a barbed wire fence along its more than 100-mile border with Serbia in a bid to prevent migrants crossing illegally as they make their way north.

Government spokesman Zoltan Kovacs told CNN's "The World Right Now" the lack of documentation was a key issue for authorities.

"That's one major problem we have to face: that these people, at least to the Hungarian border, came without papers, or got rid of papers, and at the end of the day it's basically impossible to determine whether they are telling the truth that they are coming from Syria or other war zones," he said.

In an earlier statement, Kovacs called on Germany to clarify its position on allowing Syrian refugees into Germany from Hungary, "calling for order and legality to be restored at the borders of the European Union."

Refugees flood into Germany

Even with the bottleneck in Hungary, migrants were still making their way to Germany , just not many by train.

Germany's government said last month it expected up to 800,000 asylum seekers to come this year -- four times more than in 2014.

At Munich's main station, though, the arrival of refugees slowed to a trickle because of the restrictions in Budapest.

Photos: 24 hours of Europe's migration crisis Photos: 24 hours of Europe's migration crisis Migrants wear identification bracelets aboard a Norwegian ship during a search-and-rescue mission off the Libyan coast on Tuesday, September 1. Europe is in the midst of a migration crisis. Desperate men and women, often with children in tow, are fleeing wars and poverty to find a better life on the continent. But their voyages, both on land and on sea, can be dangerous and sometimes deadly. Hide Caption 1 of 13 Photos: 24 hours of Europe's migration crisis Hide Caption 2 of 13 Photos: 24 hours of Europe's migration crisis The body of a drowned migrant remains on a shore in Zuwara, Libya. In the last week , two boats carrying hundreds of people capsized near Zuwara, killing at least 84, officials said. Hide Caption 3 of 13 Photos: 24 hours of Europe's migration crisis A migrant shouts at Macedonian police officers while trying to cross Greece's northern border into Macedonia. About 1,500 migrants were waiting to cross the border. Hide Caption 4 of 13 Photos: 24 hours of Europe's migration crisis A young girl from Iraq waits for registration with her family outside the main station in Munich, Germany. Hundreds of migrants are arriving every day. Hide Caption 5 of 13 Photos: 24 hours of Europe's migration crisis Migrants sit outside the main station in Munich as they wait for their chance to register. Hide Caption 6 of 13 Photos: 24 hours of Europe's migration crisis A group of refugees continues its journey through Hungary after crossing the Serbian border. Hide Caption 7 of 13 Photos: 24 hours of Europe's migration crisis Syrian refugees walk across railways tracks next to the Serbian town of Horgos to cross the border and enter Hungary. Hide Caption 8 of 13 Photos: 24 hours of Europe's migration crisis Migrants sit in a police bus in Pocking, Germany, after making their way to the country from Austria. Hide Caption 9 of 13 Photos: 24 hours of Europe's migration crisis Migrants from Afghanistan drink tea at a Paris school where they found shelter. Hide Caption 10 of 13 Photos: 24 hours of Europe's migration crisis A migrant sleeps on a sidewalk in Berlin near the reception center for refugees and asylum seekers. Hide Caption 11 of 13 Photos: 24 hours of Europe's migration crisis Migrants walk on a rocky beach in Ventimiglia, Italy, where they set up camp near the French border. Hide Caption 12 of 13 Photos: 24 hours of Europe's migration crisis Refugee children sleep near a railway station in Budapest, Hungary. Hide Caption 13 of 13

"It was so hard for us. It took a very long time," said a woman from Afghanistan. "Especially in Hungary, it was very difficult to get through Hungary. We had almost no food and water."

German Chancellor Angela Merkel said Monday that her country -- where some are opposed to taking in asylum seekers -- must show "flexibility" when it comes to dealing with the crisis.

European Union member states agreed in July to take in more than 32,000 migrants to ease the burden on Italy and Greece, where the largest numbers have arrived. Another 8,000 should be allocated by the end of the year, said the European Union's commissioner for migration.

Varying sentiments

For the most part, the German public supports Merkel.

Local football clubs hoisted welcome banners over the weekend. Villages held "refugee welcome" parties for the newcomers. And a recent news poll estimated that 60% back Merkel's warm welcome.

Some other countries are just as welcoming.

In Reykjavik, Icelanders called on the government to accept more refugees from Syria.

Others are reluctant.

One of them is Slovakia, which said last month that it only wanted to take in Christians because it has only a tiny Muslim community and it would be hard for new Muslim arrivals to integrate.

Czech authorities said they have started to remove migrants from trains.