COLUMBUS, Ohio — Six years after former President Donald J. Trump paved his way to the White House on nativist and xenophobic appeals to white voters, the 2,000-mile dividing line between Mexico and the United States has once again become a fixation of the Republican Party.But the resurgence of the issue on the right has come with a new twist: Republican leaders and candidates are increasingly claiming without basis that unauthorized immigrants are gaining access to the ballot box.Voter fraud is exceptionally rare, and allegations that widespread numbers of undocumented immigrants are voting have been repeatedly discredited. Yet that fabricated message — capitalizing on a concocted threat to advance Mr. Trump’s broader lie of stolen elections — is now finding receptive audiences in more than a dozen states across the country, including several far from the U.S.-Mexico border.In Macomb County, Mich., where Republicans are fiercely split between those who want to investigate the 2020 election and those who want to move on, many voters at the county G.O.P. convention this month said they feared that immigrants were entering the country illegally, not just to steal jobs but also to steal votes by casting fraudulent ballots for Democrats.“I don’t want them coming into red states and turning them blue,” said Mark Checkeroski, a former chief engineer of a hospital — though data from the 2020 election showed that many places with larger immigrant populations instead took a turn to the right.Tough talk on illegal immigration and border security has long been a staple of American politics. Both Republicans and Democrats — especially the G.O.P. in recent years — have historically played into bigoted tropes that conflate illegal immigration and crime and that portray Latinos and Asian Americans as perpetual foreigners in their own country or, worse, an economic threat.But the leap from unsecure borders to unsecure elections is newer. And it is not difficult to see why some voters are making it.In Ohio, where Republicans vying in a heated Senate primary are discussing immigration in apocalyptic terms and running ads showing shadowy black-and-white surveillance video or washed-out images of border crossings, Mr. Trump whipped up fears of “open borders and horrible elections” at a rally on Saturday, calling for stricter voter ID laws and proof of citizenship at the ballot box.The campaign commercials and promos for right-wing documentaries that played on huge television screens before Mr. Trump’s speech seemed to alternate between lies that the 2020 election was stolen from him and overblown claims blaming unauthorized immigrants for crime. Speakers in one trailer for a film by Dinesh D’Souza, the conservative author and filmmaker Mr. Trump pardoned for making illegal campaign contributions, denounced “voter trafficking,” compared the work of what appeared to be voter outreach groups to the “Mexican mafia” and referred to people conveying mail-in ballots to drop boxes as “mules.”It is legal in some states for third parties, like family members or community groups, to drop off completed ballots — a practice that became vital for many during the pandemic.Yet the messages seemed tailor-made for rally attendees like Alicia Cline, 40, who said she believed that Democrats in power were using the border crisis to gin up votes. “The last election was already stolen,” said Ms. Cline, a horticulturist from Columbus. “The establishment is, I think, using the people that are rushing over the borders in order to support themselves and get more votes for themselves.”Alicia, left, and Cindi Cline at former President Donald Trump’s “Save America” rally last week in Delaware, Ohio.Maddie McGarvey for The New York TimesThe latest fear-mongering about immigrants supposedly stealing votes is just one line of attack among many, as Republicans have made immigration a focal point in the midterms and Republican governors face off with the Biden administration over what they paint as dire conditions at the border.Last week, governors from 26 states unveiled “a border strike force” to share intelligence and combat drug trafficking as the Biden administration has said it plans to lift a Trump-era rule that has allowed federal immigration officials to turn away or immediately deport asylum seekers and migrants.A Guide to the 2022 Midterm ElectionsMidterms Begin: The 2022 election season is underway. See the full primary calendar and a detailed state-by-state breakdown.In the Senate: Democrats have a razor-thin margin that could be upended with a single loss. Here are the four incumbents most at risk.In the House: Republicans and Democrats are seeking to gain an edge through redistricting and gerrymandering, though this year’s map is poised to be surprisingly fairGovernors’ Races: Georgia’s contest will be at the center of the political universe, but there are several important races across the country.Key Issues: Inflation, the pandemic, abortion and voting rights are expected to be among this election cycle’s defining topics.And in Washington Thursday, Republicans on Capitol Hill previewed their midterm plan of attack on the administration’s immigration policies, trying to make the homeland security secretary, Alejandro N. Mayorkas, accept blame for a historic spike in migration across the border.Jane Timken, a U.S. Senate candidate and former chairwoman of the Ohio Republican Party, said the border with Mexico loomed large for Ohioans because many saw the state’s drug and crime problems as emanating from there. “Almost every state is now a border state,” she said.Some G.O.P. strategists warn that the focus on immigration could backfire and haunt the party as the nation grows more diverse. But political scientists and historians say Republicans’ harnessing of the unease stirred by demographic shifts and a two-year-old pandemic could mobilize their most ardent voters.“When we feel so much anxiety, that is the moment when xenophobic, anti-immigrant sentiment can flourish,” said Geraldo L. Cadava, a historian of Latinos in the United States and associate professor at Northwestern University.Few races nationwide capture the dynamics of the issue like the G.O.P. Senate primary in Ohio. Contenders there are taking after Mr. Trump, who, in 2016, tried to blame illegal immigration and Mexican drug cartels for the deadly opioid crisis.An ad for Ms. Timken opens with grainy footage over ominous music, showing hooded men carrying packages presumed to be filled with drugs across the border, until Ms. Timken appears in broad daylight along the rusty steel slats of the border wall in McAllen, Texas.An advertisement released by Jane Timken, a Republican candidate for U.S. Senate from Ohio, showed her at the Mexican border wall encouraging border security and raising fears of drug cartels.Jane Timken for U.S. SenateMs. Timken said she understood the state needed immigrant workers, citing her Irish immigrant parents, but said people still must cross the border legally. And Mike Gibbons, a financier at the top of several Ohio polls, said insisting on law and order was not xenophobic. “You don’t hate immigrants if you tell that immigrant they have to come here under the law,” he said.But across this state in the nation’s industrial belt, anti-immigrant sentiment tends to run as deep as the scars of the drug epidemic.Anger and resentment toward foreigners started building as manufacturing companies closed factories and shipped jobs overseas. The opioid crisis added to the devastation as pharmaceutical companies and unscrupulous doctors profited from pain medications.But with the shuttering of “pain clinics,” federal and local law enforcement officials say, Mexican criminal organizations have stepped in. In Ohio, the groups move large amounts of meth and fentanyl, often in counterfeit pills, along Route 71, which crosses the state through Columbus. Statewide overdose rates remain among the nation’s highest.An ad for suboxone, a medication used to treat opioid addiction, hanging on a building. For the past three years, Ohio has remained among the 10 states with highest rates of drug overdoses, according to federal data.Spencer Platt/Getty ImagesJ.D. Vance, the “Hillbilly Elegy” author whom Mr. Trump endorsed, goes right at those scars, telling voters in one ad that he nearly lost his mother, an addict, to “the poison coming across our border.”Republicans like Mr. Vance argue that they are being unfairly attacked for raising legitimate concerns, pointing to enormous drug seizures and a rise in border apprehensions that, last June, reached a 20-year high.Ohio immigrant-rights lawyers and advocates say Republicans are wrongly framing a public health emergency as a national security problem and contributing to bias against Latinos and immigrants regardless of their citizenship.The G.O.P. critique, they say, is also detached from reality: Many if not most immigrants who reach Ohio have been processed by federal immigration agencies. Many are asylum seekers and refugees, and an increasing number arrive on work visas.Angela Plummer, executive director of the nonprofit Community Refugee and Immigration Services, called Republican Senate candidates’ characterizations of immigrants a disturbing flashback to Mr. Trump’s 2015 campaign rhetoric. “It is good to have politicians with different immigration platforms, but not ones that stray into racism and hurtful, harmful accusations.”In the same campaign ad, Mr. Vance goes on to say that Mr. Biden’s immigration policy also meant “more Democrat voters pouring into this country” — explicitly asserting that unauthorized immigrants are crossing over and gaining access to the ballot to support the left.Mr. Trump himself made that false claim in 2017, asserting without evidence that between three million and five million unauthorized immigrants had voted for Hillary Clinton. But the idea that immigrants, and Latinos specifically, are illegally entering the country to vote Democratic has been a fringe right-wing trope for years, said Mike Madrid, a Republican political consultant and co-founder of the Lincoln Project.The difference is that purveyors of the idea have become much more “brazen and overt,” he said. “It is all part of this sense of an invasion and a lost America and that Democrats are trying to steal elections.”Rhetoric on immigration started heating up last year amid an influx of asylum seekers and migrants from Haiti, Guatemala and Honduras. In Texas, Gov. Greg Abbott and local officials described illegal immigration as an “invasion” as Mr. Abbott unveiled plans to finish Mr. Trump’s border wall.It has only intensified with the midterm campaign season. Since January, Republican candidates in 18 states have run ads mentioning the border and slamming illegal immigration, including Wisconsin, Pennsylvania and Michigan, according to AdImpact, which tracks ad spending. In the same period in 2018, that number was only six, and most of the ads ran in Texas.At least one ad warns of an “invasion,” and others carry echoes of the “great replacement” trope, a racist conspiracy theory falsely contending that elites are using Black and brown immigrants to replace white people in the United States.In Alabama, a re-election ad for Gov. Kay Ivey shows a photo of Latinos at a border crossing wearing white T-shirts with the Biden campaign logo and the words, “Please let us in.” If Mr. Biden continues “shipping” unauthorized migrants into the United States, Americans could soon be forced to learn Spanish, Ms. Ivey says, adding: “No way, José.”An Ivey spokeswoman dismissed as “absurd” suggestions that the ad played into fears of replacement or perpetuated bias against Latinos or immigrants.Heavy-handed anti-immigrant appeals haven’t always worked. Mr. Trump’s attempts to stir fears over caravans of Central American immigrants making their way north largely failed as a strategy for Republicans in the 2018 midterms.But Democrats then had a punching bag in Mr. Trump’s policy of separating migrant families at the border, which sparked international outcry. This cycle, Democrats themselves are sharply divided on immigration, leaving them either on defense or avoiding the subject altogether.Republicans like the Ohio Republican Senate candidate J.D. Vance argue liberals are calling conservatives racist for raising legitimate concerns about drug seizures.Brian Kaiser for The New York TimesThat said, some Republican voters continue to press candidates for more than just new reasons to fear immigration, and the subjects of those fears can turn out to be far less sinister than the faceless migrants depicted in grainy campaign ads.At a campaign stop at a brewery in Hilliard, Ohio, Bryan Mandzak, 53, a factory manager, asked Mr. Vance how he planned to address what he called a broken immigration system that provided workers few paths to legal status. He said he himself had seen “vanloads of Hispanics” arriving at a hotel in Marysville, about 20 miles northwest of Hilliard, but explained that they had been brought in to run an automotive plant that was hurting for employees.As it happened, white vans were indeed picking up Hispanic workers at the hotel in Marysville, for factory shifts ending at 2 a.m. But the workers were mostly American-born citizens like Moises Garza, who said he had applied on Facebook, moved from Texas and was enjoying decent pay, transportation and free lodgings.In between bites of syrupy waffles a few hours after a Friday-night shift assembling tires, Mr. Garza, who is originally from upstate New York, said he wasn’t following the Senate race and shrugged off being mistaken for an immigrant.He had two days to rest up and explore Columbus. On Monday, he would be back at work. More