BUDAPEST, Hungary — Thousands of exhausted migrants reached their dream destinations of Germany and Austria on Saturday, completing epic journeys by boat, bus, train and foot to escape war and poverty.

Before dawn, they clambered off a fleet of Hungarian buses at the Austrian border to find a warm welcome from charity workers offering beds and hot tea. Within a few more hours of rapid-fire aid, many found themselves whisked by train to the Austrian capital, Vienna, and the southern German city of Munich.

The surprise overnight effort eased immediate pressure on Hungary, which has struggled to manage the flow of thousands of migrants arriving daily from non-EU member Serbia. But officials warned that the human tide south of Hungary still was rising, and more westward-bound travelers arrived in Budapest within hours of the mass evacuation of the capital’s central rail station.

The apparent futility of stopping the migrants’ progress west was underscored when Hungary announced Saturday that its bus service to the border had finished and would not be repeated. Almost immediately two groups of migrants hit the pavement to start walking to the border: about 200 who walked out of an open-door refugee camp near the city of Gyor, and about 300 who left Budapest’s central Keleti train station, the epicenter of Hungary’s recent migrant crisis.

About 4,000 migrants crossed into Austria from Hungary by mid-morning, according to Austrian police spokesman Helmut Marban. Vienna city official Roman Hahslinger said 2,300 had arrived in Vienna by midday. And officials in both Austria and Germany said the unregulated flow of migrants Saturday from Hungary meant that up to 10,000 might cross by nightfall.

Hungary’s nationalist government spent most of the week trying to force migrants to report to government-run refugee centers, but thousands refused and demanded free passage chiefly to Germany.

After a three-day standoff with police, thousands marched west Friday from the Keleti train station along Hungary’s major motorway and camped overnight in the rain by the roadside. Hundreds more broke through police lines at a train station in the western town of Bicske, where police were trying to take them to a refugee camp, and blocked the main rail line as they, too, marched west.

Austria and Germany made the breakthrough possible by announcing they would take responsibility for the mass of humanity that was already on the move west or camped out in their thousands at Keleti. Hungary on Tuesday had suspended train services from that station to Austria and Germany, compounding the buildup there in a futile bid to try to make the visitors file asylum papers in Hungary.