Police gathered at the side of the track as the trains abruptly stopped at Bicske.
A CNN crew on one of the trains said the families -- who boarded hoping to travel to Austria or ultimately Germany -- were refusing to get off despite suffocating heat and limited food and water.
Some youths and men were holding on to the handles of train cars in case police attempted to board and remove them forcibly. Meanwhile, parents fanned their crying children in an attempt to cool them down.
Tents and desks had been set up near the station in what the migrants feared was a relocation camp to transfer them to a nearby refugee center.
The trains had left Budapest, destination unknown, after the city's main Keleti station -- packed with weary migrants and refugees who've been waiting for days to travel onward to Western Europe -- reopened in the morning.
Only domestic trains are leaving the station, Hungarian government spokesman Zoltan Kovacs told CNN earlier.
U.N. official: European leaders have explaining to do
In recent days, Keleti station has become a focal point of the crisis engulfing parts of Europe as an unprecedented wave of people -- mostly refugees fleeing conflict in Syria, Iraq and Afghanistan -- seek to reach Northern and Western Europe.
Most every European country has been affected in some way. And leaders and citizens around the continent, for the most part, have adopted positions that they hope will deal with the crisis.
Hungary is more a transit point than an end destination on a long journey north from Greece to wealthier nations such as Austria and Germany, where they hope to claim asylum. Still, it's been inundated. U.N. refugee agency spokesman Babar Baloch said there have been 140,000 asylum applications filed in Hungary since January, with about 2,000 new arrivals daily.
Previously, these migrants were allowed to move on, according to Baloch. But not anymore, with Kovacs saying these migrants are free to travel to holding camps in Hungary, but won't be able to board an international train service.
"We don't know," said Baloch, the U.N. spokesman. "That needs to be answered by European leaders."
Hungarian authorities have said that under EU legislation, they can't allow people to travel without the proper documentation -- a valid passport, a ticket and any necessary visas.
Hungarian railway operator MAV said Thursday it had decided not to run direct trains from Budapest toward Western Europe "for safety reasons." International tickets will be accepted on domestic trains, it said.
Migrants hesitant to head to camps
There was wide unease among the families on the train boarded by CNN's crew as it pulled away, but many said they just couldn't stay at the Keleti station any longer.
Few want to go to the migrant camps set up by authorities, however, fearing that once there they may not be able to continue on to other European Union states -- and that they won't be treated with respect there.
Hungarian authorities said the camps provide shelter, food and water to those in need. But a number of migrants have said they were traumatized by their treatment at camps on the country's border with Serbia.
Some recounted a shortage of water and said that when it was provided, bottles were thrown into the crowds of people, forcing them to scramble like animals to get some.
Members of one Syrian family on the train told CNN they had been on the road for two months, having fled war at home, and are hungry, dirty and running out of money.
The dramatic scenes in Hungary come a day after the photo of a toddler's lifeless body
-- one of 12 people who drowned off Turkey and washed up on a beach -- dominated headlines around Europe.
The 3-year-old boy has been identified as Aylan Kurdi, a Syrian Kurd whose family was trying to find sanctuary in Canada. The boy's brother, Galip, and their mother, Rehen, also died, according to Fin Donnelly, a member of the Canadian Parliament.
Prime Minister: 'Don't criticize Hungary'
On Thursday, Hungarian Prime Minister Viktor Orban was meeting with other EU members to figure out how to cope with the emergency.
Speaking to reporters in Brussels, Belgium, alongside European Parliament President Martin Schulz, Orban said the situation was not of his country's making.
"The problem is not a European problem; the problem is a German problem," he said.
Germany's government said last month it expected up to 800,000 asylum seekers to come this year -- four times more than in 2014. But, Orban said, German Chancellor Angela Merkel has said that they must be registered before leaving Hungary.
"All of them would like to go to Germany; our job is only to register them," Orban said.
Without strict border controls, EU migrant quotas are "an invitation" for migrants to come, he said. "Turkey is a safe country, why don't you stay there?" he said, adding that migrants who reach Serbia should also stay put.
Hungary's right-wing government has faced criticism for 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.
But Orban said his country was just trying to enforce EU rules and that new measures would be in place shortly. "Don't criticize Hungary for what is being done. Let Hungary do the job as it is written in the European regulations," he said.
Schulz warned the EU must work together. "This is a crucial moment for the European Union," he said. "A deeper split of the union is a risk."
While European leaders struggle to come up with a coherent plan, the men, women and children caught up in the crisis continue to suffer.
On Wednesday, some refugees in Budapest held up scraps of paper: "Help Syrians," they read. "Babies are tired."
"We hope you will save us," a Syrian refugee named Houriye told CNN. "I beg you to save us."
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.
EU foreign policy chief urges 'united' action
France and Germany are sending common proposals to the EU to organize the welcome and fair distribution of refugees in Europe, the French President said in a statement.
French President François Hollande and Germany's Merkel stressed the need to reinforce the European asylum system to ensure the return of migrants who are in an irregular situation to their country of origin and to support countries of origin and transit, the Élysée Palace said.
Speaking in Paris, Hollande said it was "time to act" to prevent more tragedies such as the death of Aylan Kurdi -- and that some countries not doing enough must step up and shoulder their share of the burden.
The EU's foreign policy chief, Federica Mogherini, urged "united" action Wednesday and said the issue would be part of the next EU foreign ministers' meeting on Friday and Saturday. The EU Commission was meeting to prepare new proposals.
EU Council President Donald Tusk warned of a divide between Eastern and Western European nations in the way they handle the situation.
"Some member states are thinking about containing the wave of migration, symbolized by the controversial Hungarian fence. Others expect greater solidarity in advocating ... a so-called obligatory basis for quotas," he said.
Joyful refugees reach Germany
Germany's government has been more accepting of asylum seekers than some, but political leaders there have had to contend with xenophobic protests.
The German Chancellor said Monday that her country must show "flexibility" when it comes to dealing with the crisis.
Caught in the middle of the confusion are the desperate men and women, with children in tow, who have fled wars in Syria and Iraq to embark on fraught, sometimes deadly voyages by land or across the Mediterranean Sea.
Tired, hungry, weak from the heat, those who arrived in Munich this week spoke of harrowing journeys -- and the joy of finally making it to Germany.
"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."
Border controls reinstated
EU rules usually allow people with proper documentation to travel through much of the EU without border checks -- but the current influx of migrants has led to extraordinary measures in some places.
Italian authorities on Wednesday said they were temporarily reinstating border controls at the Italian-Austrian line in the Alto Adige region after Bavarian authorities in Germany requested them to do so because they are "overwhelmed" by the influx of migrants, according to a statement 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.
Czech authorities, meanwhile, said they have started to remove migrants from trains.
In the United Kingdom, Prime Minister David Cameron has come under new pressure to offer shelter to more refugees from the Middle East after saying the best policy was to focus on bringing peace to the region.
The Council of Europe commissioner for human rights, Nils Muiznieks, said that Cameron's position "seriously concerned" him.
"The truth is that at the moment the UK is doing much less than other European countries, like Germany or Sweden, which give refuge to thousands of Syrians," Muiznieks said in a statement.
An online petition
calling for the UK Parliament to accept more asylum seekers has passed the 100,000 mark required to ensure debate.
"We can't allow refugees who have risked their lives to escape horrendous conflict and violence to be left living in dire, unsafe and inhumane conditions in Europe. We must help," the petition states.
In Iceland, citizens have also called online for their government to take in more Syrian refugees.