Groups of the refugees earlier had left a rail station and stopped train and begun walking toward Austria. Also: The father of a 3-year-old Syrian boy who drowned in the Aegean Sea buries his family.
More than 1,000 weary Syrian refugees were greeted with food and applause at the Austrian border after arriving by busloads from a long, chaotic journey through Hungary.
Police in Budapest lifted a two-day blockade of the Keleti station, allowing migrants to enter. But some inadvertently boarded trains bound for a refugee camp outside the capital.
Prime Minister Viktor Orban has adopted staunchly anti-migrant positions to try and win back supporters from the far right, which is making significant gains.
In a speech last week, James Comey had linked the two countries to the killing of Jews during the Holocaust. They have both said the killings occurred when they were occupied by the Nazis.
Hungary's national gas operator says the "indefinite" cutoff is over domestic supply concerns, but the move comes just days after the CEO of Russia's Gazprom monopoly visited Budapest.