Paul C. Pauley views himself as a middle-of-the-road Republican — who also just happens to believe in one of the most pernicious far-right conspiracy theories about illegal border crossings: that Democrats are bringing undocumented immigrants into the country to vote for their party.

“I don’t think they’re going to get that vote this year,” said Mr. Pauley, who sells evergreens on a family farm near Warren, Ohio. “But four years from now? Eight years from now?”

Former President Donald J. Trump and his running mate, Senator JD Vance of Ohio, are pushing the idea to mobilize supporters based on fear of what they call a foreign “invasion.” But demographers and population studies experts say that there is no evidence for the claim. It also strains even the imagination, envisioning Democrats in Washington circumventing border rules, officials and infrastructure to allow undocumented immigrants into the country, help them settle and then cast ballots, legally or not.

Election, court and law enforcement officials who have monitored reports of voter fraud have found no proof of widespread fraud, and they have said that the number of noncitizens who have wrongfully cast ballots is minuscule. It is a crime for noncitizens to attempt to vote in federal elections.

More than a dozen cities and towns, mostly in deep-blue areas, allow foreign nationals to vote in local elections regardless of their immigration status, which state and local leaders say is warranted because unauthorized immigrants pay taxes at levels comparable to those of citizens and strengthen their economies. Many immigrants wait for years to become naturalized and do not register to vote or cast ballots once they do.