Inspired by phone calls and Facebook posts from friends hiking through the Balkans, crossing into Germany or simply touching dry land in Greece, people from countries long plagued by war and instability say they are seizing a pivotal moment.

“This is a golden opportunity,” said Osama Ahmed, 27 years old, who lined up Sunday at Baghdad International Airport, heading for Greece via Turkey with five friends. “It’s totally nonsense to stay in Iraq when there is a chance to go.”…

Baghdad travel agents report surging demand for plane tickets, prompting airlines to add three more daily flights to Istanbul—on top of five packed flights a day already. Some Iraqis are forgoing resettlement applications for the U.S., a process that takes years.

“We got many phone calls and emails from friends already abroad telling us to leave Iraq now—immediately—since the European authorities are being easy on migrants,” said Mr. Ahmed, who said his plan is to reach Belgium.