Refugees have been transported to a registration centre by Hungarian authorities hoping to defuse tensions at the country's border with Serbia.
A steady stream of people continue to approach Hungary, which they see as their gateway into the European Union.
Hungary has made frantic and confused efforts to control the tide of migrants entering the country as they try to reach Germany, leaving many trapped for days outside the border village of Roszke.
Prime Minister Viktor Orban has announced fresh efforts to complete a fence to keep the refugees out.
The UN refugee agency said it is concerned about the lack of proper reception facilities in the border area and that humanitarian aid needs to be stepped up there.
Babar Baloch told The Associated Press that "the border police are not trained to deal with the refugees".
"When people come in you need to receive them properly. There are women and children and they are just kept in the open. Temperatures are going down as well so we need to have a system where these people receive proper care," he said.
Earlier in the day, refugees anxious to pass through Hungary who were frustrated at being trapped behind a police line began pushing and police shoved back. One man was hurt in a stampede.
Despite the efforts of volunteers offering water and some clothes, there were few amenities at the border. The area was strewn with rubbish and more people could be seen walking along rail tracks in Serbia on their way to Hungary. Many of the travellers had slept outdoors in a field during a cold night.
At Budapest's Keleti train station, migrants were being allowed to board trains bound for Austria and Germany. In many cases, they were segregated from other passengers and told they could only enter certain carriages.
The queue of people waiting to board a train to the West was growing, with about 300 people waiting for the next train on Tuesday afternoon.
Almost all of those passing through Hungary are hoping to reach Germany or other Western European countries with generous welfare benefits and open asylum regulations. Almost none wish to remain in Hungary, which is seen as unwelcoming to asylum seekers and does not have the same economic opportunities as Germany.
Around the EU, debate continued over what each country should do. Hungary and other former east bloc countries have resisted accepting refugees, but Germany has thrown its weight behind a scheme to set a quota for each of the 28 EU nations.
Austrian Chancellor Werner Faymann said countries opposed to taking in refugees under an EU-wide quota system should suffer financial penalties.
In Geneva, UN official Peter Sutherland said it is "not enough" for countries like the United States and wealthy Persian Gulf states to give money to help Syrian refugees - they must take them in, too.
"Buying your way out of this is not satisfactory," Mr Sutherland said, adding that UN agencies are well short of their funding needs for meeting the crisis.
Some 7,000 migrants arrived in Vienna between Monday and Tuesday and "almost all of them continued on to Germany," Austrian police spokesman, Patrick Maierhofer, said.
Poland, the largest of the eastern members, has so far agreed to accept 2,000 refugees and has been criticised for lacking solidarity with Germany, which has said it expects to take in 800,000 asylum-seekers this year and is able to absorb half a million per year for a few years.
Poland's defence minister and deputy prime minister, Tomasz Siemoniak, faulted other European nations for lacking a thought-out strategy to handle the crisis and for pushing for EU states to accept quotas of refugees.
He called that a "road to nowhere" which would only encourage more people to come. He also said that Germany should not feel it has the right to teach Poland about solidarity.
"I think that our position is filled with solidarity and with the vision that we will not solve the problem with such summary, hasty decisions," Mr Siemoniak said.
In Hungary, Mr Orban was quoted in the pro-government Magyar Idok (Hungarian Times) daily newspaper as saying he was persuaded to deploy more workers to finish the fence along the border after an unannounced inspection of the barrier on Monday.
The 4-metre high fence along the 109-mile border with Serbia was supposed to be completed by August 31 but is facing delays. Defence minister Csaba Hende, who was overseeing the construction being done mostly by soldiers, resigned on Monday after Mr Orban's visit because of the slow progress.
Several coils of razor wire are stretched out along the whole border but it has been regularly breached by migrants, who usually crawl under it. The higher barrier is up only on some sections.
Hungarian police said they have detained more than 169,000 migrants this year, including 2,706 on Monday.
In that same period, the Migration Office has received nearly 158,000 asylum requests.
In Greece, the coast guard said its patrol vessels picked up nearly 500 migrants in 11 search and rescue missions over the previous 24 hours in the eastern Aegean Sea.
The people, whose nationalities were not immediately clear, were found in small boats near the islands of Lesbos - which accounts for nearly one in two migrant arrivals in Greece - Samos, Kos and the islet of Agathonissi.
More than 15,000 refugees and migrants are stranded on Lesbos, awaiting screening before they can board a ferry to the Greek mainland - from where they head north through Macedonia, Serbia and Hungary to seek asylum in more prosperous European countries.