The children were taken to hospital in the town of Braunau am Inn on August 28 suffering from severe dehydration.
Their discovery came a day after 71 bodies, thought to be migrants, were found on a dumped truck in Austria.
Several European countries have called for urgent talks on the migrant crisis.
Austrian police said they stopped the minivan in Braunau, which sits on Austria’s border with Germany, on August 28 and arrested its Romanian driver.
The children – two girls and a boy aged between one and five years old – were said to have been crammed in the back along with other migrants from Syria, Afghanistan and Bangladesh.
Police said they were critically ill and almost unconscious when they were found.
The children and their families disappeared from the hospital at some point on August 29.
Authorities believe they may have tried to cross the border into Germany, rather than face deportation back to Hungary.
Separately on August 30, Hungarian police said they had arrested a fifth man over the deaths of the 71 people who were found in the abandoned truck in Austria on August 27.
The man is the fourth Bulgarian to be held over the find near the Hungarian border. The other man is Afghan. Authorities believe the men are low-level members of a human trafficking gang.
Officials said the 59 men, eight women and four children – thought to be mainly Syrians – had probably died of suffocation two days earlier.
It is the latest in a series of tragic events as more and more migrants attempt to reach Europe by land or by sea. A record number of 107,500 migrants crossed the EU’s borders last month.
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