ROSZKE, Hungary —Hundreds of angry and frustrated asylum-seekers broke through police lines Monday near Hungary’s southern border with Serbia and began marching north toward Budapest, while Britain and France pledged to take in tens of thousands more refugees to try to ease the crisis.

As European leaders debated how to share responsibility for the more than 340,000 people from the Middle East, Africa and Asia who are already seeking refuge, Germany promised to spend billions of euros in extra aid for those already there and those yet to arrive. France weighed whether increased airstrikes against Islamic State militants would help to stem the flow of those fleeing Syria.

A Syrian man and his child board a ferry traveling to Athens, at the port of Lesbos Island, Greece, on Monday. The island of 100,000 residents has been transformed by the sudden new population of 20,000 migrants. The Associated Press

Syrian refugees wait at the port of Lesbos island, Greece, to board a ferry traveling to Athens on Monday. The island of some 100,000 residents has been transformed by the sudden new population of some 20,000 migrants, mostly from Syria, Iraq and Afghanistan. The Associated Press

A migrant child holds cuddly toys at the Westbahnhof station in Vienna, Austria, Monday, Sept. 7. The Associated Press

A refugee carries a picture of German Chancellor Angela Merkel as he arrives at the main train station in Munich, Germany, Saturday, Sept. 5. Hundreds of refugees arrived in various trains to get first registration as asylum seekers in Germany. The Associated Press

A woman gives some euro coins to a refugee child, who has arrived by train at the central station in Munich, southern Germany on Saturday, Sept. 5. Since Saturday more than 7,000 Arab and Asian asylum seekers surged across Hungary’s western border into Austria and Germany following the latest erratic policy turn by Hungary’s immigrant-averse government. The Associated Press

But the Hungarian prime minister scoffed at a proposed quota system for refugees in the 28-member European Union, saying it won’t work unless Europe first secures its borders.

Hungary’s inability to control the flow of people across its southern border with Serbia was on graphic display Monday. Crowds who had grown tired of waiting for buses at Hungary’s first migrant holding center near the border village of Roszke tore down flimsy police tape, advanced down a country road and walked around and straight through rows of police trying to block them.

Police shoved individual migrants and fired jets of pepper spray, but it had little effect as about half of the 500-strong crowd reached the M5 highway that connects Serbia and Hungary. They headed north along the shoulder, raising their arms and chanting “Germany! Germany!”

MOST SHUN BUS TO REFUGEE CAMP

Police merely walked beside them as a lone helicopter monitored the marchers’ progress north as darkness fell. The highway was blocked for nearly 30 miles as a precaution. A few hours later, as the marchers paused by the roadside to try to sleep in the cold on the pavement, police delivered buses and requested they board for delivery to a refugee camp. Most refused.

The northward march mirrored Friday’s surge of people down Budapest’s motorway toward Austria, which forced Hungary to concede defeat and bus thousands to the Austrian border. Germany’s rail company said Monday it had carried 22,000 asylum-seekers over the weekend on more than 100 trains, a number boosted by the fact that Hungary again has dropped visa checks on foreigners buying train tickets for the wealthier countries to the west, particularly Germany.

Following an overnight Cabinet meeting, Germany said it would set aside $6.6 billion to boost aid for asylum-seekers and hire 3,000 more federal police. It also planned to make it easier to build refugee housing and for non-German speakers to hold jobs.

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