ROSZKE, Hungary — Clambering over the razor-wire fence or scuttling under it, migrants surged Wednesday across the Serbian border into Hungary. Then they jostled to formally enter the country so they could quickly leave it, heading toward more prosperous European Union nations on a desperate quest to escape war and poverty.

In Roszke, a Hungarian border town, migrants mostly from Syria, Iraq and Afghanistan requesting asylum were being processed by authorities. Police used tear gas to break up a brief scuffle involving about 200 migrants, saying the migrants were growing impatient with delays in the registration process.