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.

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

Hungary’s inability to control the flow of people across its southern border with Serbia was on graphic display Monday.