Thousands of migrants have flooded into Austria, seeking refuge after shuttling for days between bordering countries that were unable or unwilling to offer them shelter.

Austrian police said some 6,700 people travelled to the central European country from Hungary after being trapped on Friday in a vicious tug-of-war as bickering European governments shut border crossings, blocked bridges and erected new barbed-wire fences in a bid to shut down the flow.

More are expected as people continue to make their way north via Turkey and Greece after fleeing conflict and poverty in the Middle East and Africa.

On Saturday, the Greek coast guard said they failed to save a five-year-old girl found in the sea off the island of Lesbos after the boat she travelled on sank, also leaving 14 others missing.

Asylum seekers who headed westward into Croatia after being beaten back by tear gas and water cannons on the Hungarian-Serbian border just days earlier found themselves being returned to Serbia or to Hungary, after Croatia declared it could not handle the influx.

Hungary then put them on buses and sent them on to Austria. More were expected on Saturday.

Meanwhile, Hungary's military said that it is calling up 500 army reservists as the country reinforces its borders with razor-wire fences, the deployment of thousands of soldiers to the border and other tough measures.

The European Union's failure to find a unified response to the crisis left Croatia, one of the poorest countries in the EU, squeezed between the blockades thrown up by Hungary and Slovenia and the unending flood of people flowing north from Syria, Iraq and Afghanistan.

Though sympathetic to their plight, Croatian president Kolinda Grabar Kitarovic demanded that the EU step forward and take responsibility for the people in transit through this country of 4.2 million people. More than 20,000 have arrived since Wednesday.

"We're flooded, local communities are flooded, the numbers of refugees in some areas is far greater (than) the number of local residents," she told the Associated Press. "So we need to control, we need to stop the flow, we need to get reassurances from the European Union (about) what happens to these people who are already in Croatia, and those who still want to transit through Croatia."

Mindful of people crossing cornfields and forests to transit her country, Ms Kitarovic stressed further measures would be taken to secure Croatia's borders. Underscoring that Croatia itself has only recently begun to recover from the Balkan Wars of the 1990s, Ms Kitarovic said that migrants were also in danger of stepping on mines left over from the conflict.

"I will advise highly anyone to use official crossings, but we have to take further measures to ensure stability on the border, and that there are no breaches through the cornfields, or forests or any other areas that are not controlled or cleared," she said.

The thousands seeking sanctuary as doors close behind them are camping in the open, sleeping on streets, exposed to heat in the day and cold in the night.

Police in Slovenia say more than 1,000 migrants have entered the country, but hundreds more are waiting at the border as they let in only limited numbers.

As temperatures dipped overnight, hundreds of migrants at the Obrezje crossing set up tents, camping there without food and water.

Ammar Jessem, 24, a dentist from Baghdad, said the migrants had heard the border was open and were disappointed after walking through Croatia in the hopes of going north.

"We don't want to stay here. We want to go to Austria. From Austria we go everywhere," he said.

"It is a small country and we can go by walking," he added, undeterred. "It is no problem for us."