OTTAWA — The move-to-Canada memes started spreading in earnest shortly after Donald J. Trump looked increasingly likely to win Florida: “Election Night Starter Kit,” read a post on Instagram, above photos of United States passports and an Air Canada plane. Another post depicted a machine-gun-toting man riding a moose, with the words “Canadian Border Patrol Watching for Illegal Americans.”

The comedians may have been on to something. Later on Tuesday night, the website of Canada’s immigration department crashed. Social media quickly filled with speculation that it had been overloaded by Americans looking for a new country to call home.

Lisa Filipps, a spokeswoman for Citizenship and Immigration Canada, said on Wednesday that the department’s site had failed “as a result of a significant increase in the volume of traffic,” but gave no details about the source of the traffic. The site was back in operation by midmorning but responded slowly and erratically.

Such a desire to flee an election result in the United States is often expressed, but rarely borne out. On the other side of the border, Canadians reacted to Mr. Trump’s election win with concerns of their own, even anxiety, but also a whiff of pride.