By the time the Arizona Republican Party’s digital billboards urging Phoenix residents to “EAT LESS KITTENS” and “Vote Republican!” went up on Tuesday, the story they referred to had already been widely debunked.

The previous week, claims that Haitian immigrants were stealing and eating pet cats in Springfield, Ohio, had begun circulating on social media. None of the reports had been substantiated, and local officials in Springfield — a small city where, in recent years, public services have been strained by a large influx of Haitian migrants — said they had seen no evidence that any of them were true. Major conservative media outlets had given them only fleeting attention.

But while the story has so far not proved credible, it has proved meme-able — and that has given it a life far beyond the right-wing internet.

For days, images and videos of former President Donald J. Trump and cats have proliferated online, delivered with a knowing wink and an understood endorsement of Mr. Trump’s hard-line immigration message. Their over-the-top imagery gives them the feel of an inside joke. A “just kidding” is implied, allowing political figures who might otherwise have hesitated to circulate debunked material to get in on it. The reality is beside the point.

Memes have been a regular, if enigmatic, feature of American politics since the primordial days of social media. But the early months of the Trump-Harris race have offered a twist on this familiar phenomenon: More than once, memes that are mostly or completely detached from actual events have spilled off the internet into the three-dimensional reality of the campaign itself.

They are now on the billboards in Arizona and were, more significantly, on the debate stage Tuesday night, where Mr. Trump exclaimed: “The people on television say their dog was eaten by the people that went there!”