That came after police sealed off the terminal to prevent them from travelling through the European Union.
Thousands of people use Hungary as a gateway to northern Europe to claim asylum.
Meanwhile, Greece has begun transporting ferries carrying thousands of asylum seekers from its islands to the mainland.
There have been angry scenes at Keleti station in Budapest as asylum seekers are barred from boarding trains out of Hungary.
Authorities have prevented at least a thousand people from catching trains to Austria and Germany.
After nearly four thousand asylum seekers arrived in Vienna from Hungary on Tuesday, authorities stopped all trains to check tickets.
Now, only non-asylum seekers are being allowed to leave by train.
German politician Annette Groth, visiting Hungary on a fact-finding mission, has told Al Jazeera she is appalled by the situation.
“This is a complete failure of human rights in Europe, this is what I have to say here. Massive human-rights violations here. People sleeping on the streets for days and days and days. There’s hardly any water, hardly any food.”
But Hungarian government spokesman Zoltan Kovacs says his government is only applying European Union laws.
“It is impossible to come to Hungary without going through another European Union state, namely Bulgaria … or, rather, Greece and Bulgaria. And it’s completely unacceptable that the Greeks basically let these people come towards Hungary and towards the heart of Europe without any kind of registration.”
Europe is in the midst of the largest movement of people since the Second World War.
More than 300,000 people have arrived in Europe this year, fleeing from the Middle East and northern Africa.
But the European Union remains divided on how to address the issue.
Under current EU regulations, asylum seekers must remain in the first European country they enter while their protection applications are being assessed.
Germany is pushing for all EU countries to accept Syrian refugees.
Germany’s Chancellor Angela Merkel says quotas must be established to ensure a fair distribution of asylum seekers across Europe.
“This should apply in every country of the European Union. We are, of course, following the rules, but we are seeing every day how these rules are not put into practice in the same way everywhere.”
But Hungary’s Foreign Minister, Peter Szijjarto, says the quota system only encourages further exploitation of asylum seekers.
“It has been proven that some European leaders act contrary to European values and interests, as far as the handling of the challenges of migration is concerned. It became clear that policy based on mandatory quotas has failed, because it just encourages the human traffickers and immigrants to come to Europe. Meanwhile, those who are trying to make significant steps to stop the influx of migrants to Europe are facing a smear campaign based on false allegations or criticism.”
In Greece, more than 400,000 asylum seekers have been moved from the country’s small islands to the mainland.
Arriving at Athens’ main port, Piraeus, these were the reactions of the asylum seekers, most of them from Syria and Iraq.
“I want to say to Hungary, ‘Why? Why you want to stop the Syrians and the others? Why?’ We search about good life. We search about peace. We are human. We are human.”
“I wish the UN and other independent international organisations provide the help to these countries to support us.”
“Things got bad there (Syria) since four years ago maybe, the fanatic Islamic fighters and terrorism. So, I just came here seeking a chance to a better life.”