PALM BEACH, Fla. — Votes switched by Venezuelan software. Voting machines hacked by the Chinese. Checking for telltale bamboo fibers that might prove ballots had been flown in from Asia. After the 2020 election, Donald J. Trump and his allies cycled through a raft of explanations for what they claimed was the fraud that stole his rightful re-election as president, all of them debunked.

Yet on a recent evening at his Mar-a-Lago resort, there was Mr. Trump showcasing his latest election conspiracy theory, one he has been advancing for months at rallies for his favored midterm candidates.

The basic pitch is that an army of left-wing operatives stuffed drop boxes with absentee ballots — a new spin on an old allegation that voter-fraud activists call “ballot trafficking.” And while MAGA-world luminaries like Rudolph W. Giuliani, Representative Marjorie Taylor Greene and the MyPillow founder Mike Lindell filled the gilded ballroom, the former president called out two lesser-known figures sitting up front — the stars of “2000 Mules,” a documentary film promoting that ballot-trafficking theory and premiering at Mar-a-Lago that night.

“These people are true patriots,” Mr. Trump said, gesturing from the podium to the pair — a Tea Party veteran from Texas, Catherine Engelbrecht, and Gregg Phillips, her full-bearded sidekick, a longtime Republican operative — and imploring them to “stand up.”

While the early primaries have delivered a mixed verdict on the former president’s endorsements and stolen-election obsessions, polling nonetheless shows that a majority of Republicans believe the 2020 presidential election was stolen, even though vote fraud is exceedingly rare. Mr. Trump and his allies hope “2000 Mules,” now playing at several hundred theaters, will win over doubters among establishment Republicans.

Ms. Engelbrecht, the founder of True the Vote, a group that has spent years warning of the dangers of voter fraud, has criticized the earlier narratives of the 2020 election as unhelpful. “What they were putting out there was a lot of misinformation that just wasn’t true,” she said in a recent interview. “People want to believe the conspiracies in some ways.” Their film, she maintains, offers a more-serious theory.

Yet a close look at the documentary shows that it, too, is based on arguments that fall apart under scrutiny.

The film, directed by the conservative commentator Dinesh D’Souza, is based in part on an erroneous premise: that getting paid to deliver other people’s ballots is illegal not just in states like Pennsylvania and Georgia where True the Vote centered its research and where third-party delivery of ballots is not allowed in most cases, but in every state.

What’s more, the film claims, but never shows in its footage, that individual “mules” stuffed drop box after drop box. (Mr. Phillips said such footage exists, but Mr. D’Souza said it wasn’t included because “it’s not easy to tell from the images themselves that it is the same person.”) Those claims are purportedly backed up by tracking cellphone data, but the film’s methods of analysis have been pilloried in numerous fact-checks. (True the Vote declined to offer tangible proof — Mr. Phillips calls his methodology a “trade secret.”)

More broadly, Ms. Engelbrecht has said that the surge of mail-in voting in 2020 was part of a Marxist plot, aided by billionaires including George Soros and Mark Zuckerberg, to disrupt American elections, rather than a legitimate response to the coronavirus pandemic.

Mr. Phillips, whose firm OpSec does data analysis for True the Vote, is perhaps best known for making a fantastical claim in 2017 that more than three million illegal immigrants voted in the 2016 election, which was amplified by Mr. Trump but never backed up with evidence. Mr. Phillips is also an adviser to Get Georgia Right, a political action committee that received $500,000 from Mr. Trump’s Save America PAC this past March 25, the day after Mr. Phillips and Ms. Engelbrecht advanced their 2020 vote-fraud theories to a legislative committee in Wisconsin. Mr. Phillips said he had “received zero money” from Get Georgia Right, which backed Mr. Trump’s favored and failed governor-primary candidate, David Perdue.

Mr. Phillips and Ms. Engelbrecht have become controversial even within the hard-right firmament. They are embroiled in litigation with True the Vote’s largest donor, and Ms. Engelbrecht has feuded with Cleta Mitchell, a leading Trump ally and elections lawyer. John Fund, a prominent conservative journalist who was once a booster of Ms. Engelbrecht, has implored donors to shun her, according to videotape provided to The New York Times by Documented, a nonprofit news site.

“I would not give her a penny,” Mr. Fund said at a meeting of members of the Council for National Policy, a secretive group of right-wing leaders, in the summer of 2020. “She’s a good person who’s been led astray. Don’t do it.”

But Ms. Engelbrecht found support from Salem Media Group, which distributes right-wing talk radio and podcasts, including one hosted by Mr. D’Souza, who was pardoned by Mr. Trump after being convicted of campaign finance fraud. After meeting with Mr. Phillips and Ms. Engelbrecht, Salem Media spent $1.5 million to make the film and $3 million to market it, according to Mr. D’Souza. An elaborate and shadowy film set, with giant screens and flashing lights, was built to show Ms. Engelbrecht and Mr. Phillips conducting their cellphone-data analysis.

The group has not presented any evidence that the ballots themselves — as opposed to their delivery — were improper. “I want to make very clear that we’re not suggesting that the ballots that were cast were illegal ballots. What we’re saying is that the process was abused,” Ms. Engelbrecht said in Wisconsin. In an interview, she backtracked, but when asked to provide evidence of improper votes, she only pointed to previous accusations unrelated to the 2020 general election.

A repeated contention of the documentary is that getting paid to deliver other peoples’ ballots is illegal in every state. Mr. D’Souza emailed The New York Times a citation to a federal statute that outlaws getting paid to vote — and does not discuss delivering other people’s ballots. Hans von Spakovsky, a Heritage Foundation fellow, appears in the movie agreeing that the practice is outlawed nationwide, but in 2019 he wrote that it was “perfectly legal” in some states for “political guns-for-hire” to collect ballots. (Asked about the discrepancy, Mr. von Spakovsky said he believed the practice is illegal based on federal law.)

The swing states where Mr. Phillips and Ms. Engelbrecht focused their research do ban the delivery of ballots on behalf of others, with some exceptions. But elections officers in 16 other states surveyed by The Times said their states did not prohibit people getting paid to deliver a ballot. Some of those states limit how many ballots an individual can deliver, or bar campaigns from doing so.

Mr. Phillips and Ms. Engelbrecht’s case is largely built on cellphone data. A report created by the group includes an appendix that claims to list “IMEI” numbers of the tracked devices — 15-digit codes unique to each cellphone. But each entry on the list is a 20-character string of numbers and letters followed by a lot of x’s. Mr. Phillips said new IDs had been created “to obfuscate the numbers.”

The same report says the group “purchased 25 terabytes of cellphone signal data emitted by devices” in the Milwaukee area in a two-week period before the 2020 election. They claim to have isolated 107 unique devices that made “20 or more visits to drop boxes” and “multiple visits to nongovernmental organizations” that were involved in get out the vote efforts.

A number of researchers have said that while cellphone data is fairly precise, it cannot determine if someone is depositing ballots in a drop box or just passing by the area.

“It’s really, really hard to assign even what side of the street you’re on when you’re using this kind of data,” said Paul Schmitt, a research scientist and professor at the University of Southern California.

True the Vote focuses on Democrats, but in 2019 the head of the National Republican Congressional Committee vowed to be more aggressive in its use of the same practice of collecting mail-in ballots. In fact, one of the few specific allegations of vote fraud cited in the film concerns a North Carolina Republican operative who was facing ballot-tampering and obstruction-of-justice charges when >he died last month. The case led state elections officials to order the first redo of a federal election because of fraud allegations.

As True the Vote’s funding faltered in recent years, it found new adversaries among old friends. During an elections panel hosted by the Council for National Policy in the summer of 2020, Mr. Fund, a former member of The Wall Street Journal’s editorial board, said Ms. Engelbrecht had “hooked up with the wrong associates” and “gone astray.”

Ms. Mitchell, the elections lawyer and leading vote-fraud activist in her own right, was sitting next to Mr. Fund and nodding through his comments. “It’s true,” she said after he finished.

Ms. Mitchell once represented Ms. Engelbrecht in a dispute with the I.R.S. over allegations that it was targeting conservative groups. Ms. Engelbrecht said that Ms. Mitchell has been “on a rampage against me” since she fired Ms. Mitchell in 2017.

She believed the I.R.S. litigation had been “a great fund-raising vehicle” for Ms. Mitchell “and her associates. I wanted to win the case and move on and not be a cottage industry.”

Ms. Mitchell, in a text, said that her legal team spent years doing “all the heavy lifting,” but was fired after issuing a public statement, which had been standard practice. “Problem is that Catherine hates it if anyone else deals with reporters,” she said.

Soon after the 2020 election, with its funding faltering in recent years, True the Vote got a windfall $2.5 million donation from Fred Eshelman, a North Carolina entrepreneur seeking evidence to overturn the election. But the effort sputtered and Mr. Eshelman sued, claiming he had been swindled.

He lost an initial round in Texas court and is now appealing. The suit alleges Ms. Engelbrecht and Mr. Phillips were in a romantic relationship and violated Texas law related to conflicts of interest, since True the Vote directed a “substantial portion” of Mr. Eshelman’s funds to OpSec.

Asked about a personal relationship, Ms. Engelbrecht said, “You know, Gregg and I have actually talked about this and how we would answer this question. And the best answer that I think either of us are going to give is, it is totally unrelated and unimportant.”

True the Vote’s eventual focus on ballot trafficking was inspired by an Arizona investigation into ballot collection in the 2020 primary that led to indictments.

But True the Vote’s efforts have prompted little action from law enforcement. Last year, after True the Vote circulated its research in Georgia, the Georgia Bureau of Investigation said the cell data turned over, which tracked people to within 100 feet, was insufficient to act on.

“What has not been provided is any other kind of evidence that ties these cellphones to ballot harvesting,” the bureau said in a letter. “For example, there are no statements of witnesses and no names of any potential defendants to interview.” It added that while the group had said it had “a source” who could validate such findings, “despite repeated requests that source has not been provided.”