GREEN BAY, Wis. — Eric Genrich is running a full-throated campaign in support of abortion rights, reminding voters of his position at every turn and hammering his anti-abortion opponent in television ads. At a recent event, he featured an obstetrician who now commutes to a state where abortion is legal to treat patients and a local woman who traveled to Colorado to terminate a nonviable pregnancy.There’s just one inconvenient reality: Mr. Genrich is running for re-election as mayor of Green Bay, Wis., an office that has nothing to do with abortion policy.Even before the Supreme Court overturned Roe v. Wade last summer, putting back into effect a Wisconsin law from 1849 that bans nearly all abortions, the city did not have a clinic that performed the procedure, nor a health department that regulated it.Mr. Genrich is one of several candidates for municipal offices on the ballot this spring in races in Wisconsin, Chicago, St. Louis, Lincoln, Neb., and elsewhere who are making their support for abortion rights — and often their opponent’s past opposition — a centerpiece of their campaigns, even though abortion policy in all of these places is decided at the state level.Mayor Eric Genrich of Green Bay, Wis., left, has made abortion rights central to his re-election campaign. At a recent news conference, one speaker was Dr. Anna Igler, second from right, a Wisconsin obstetrician-gynecologist who traveled to Colorado for an abortion because her fetus had a severe abnormality.Kayla Wolf for The New York TimesDemocrats used a muscular defense of abortion rights to great success in the midterm elections last fall, and, if that strategy works again, they are likely to copy it next year in races at all levels of government, including in President Biden’s campaign if he seeks re-election.The focus on abortion rights in down-ballot races, however, reflects Democrats’ increased nationalization of local politics. For decades, local Republican candidates ran on issues like abortion, immigration and national security, putting them in simple terms: “A noun, a verb and 9/11,” Mr. Biden once said in describing the phenomenon.Now Democrats are doing the same on abortion in left-leaning cities, hoping to win over independent voters and some moderate Republicans.Doing so allows Democrats to avoid discussing crime rates or other less appealing campaign topics. But beyond that, they recognize and emphasize that in today’s tribal politics, the precise responsibilities of an office matter less than sending a strong signal to voters about one’s broader political loyalties.“It’s definitely not a municipal issue per se,” Mr. Genrich said in an interview. “Voters don’t care about some of these parochial distinctions between municipal boundaries. This is a city issue, a state issue, a federal issue. Some of their most important questions are, what do you stand for fundamentally?”Mr. Genrich declined repeated opportunities to explain what, precisely, the mayor of Green Bay could do about abortion in his city.Still, Republicans running for mayor find themselves doing a political tap dance, trying to de-emphasize but not disavow their opposition to abortion rights, which is not an electoral winner in Democratic cities. In Green Bay, Mr. Biden won 53 percent of the vote in 2020; last year, Gov. Tony Evers, a Democrat, took 55 percent of the city’s vote.Mr. Genrich’s opponent in Tuesday’s officially nonpartisan election, Chad Weininger, is a former state legislator who cast a series of votes to restrict abortion rights before last year’s Supreme Court ruling. Now, as television ads and campaign mail blast his stance and label him “MAGA Chad” to emphasize his Republican politics, he is trying to change the subject.Chad Weininger, who is running for mayor against Mr. Genrich in Green Bay, is a former state legislator who has opposed abortion rights in the past. Kayla Wolf for The New York Times“I’m running for mayor, I’m not debating abortion,” Mr. Weininger said. “We could have discussions about nuclear arms, but guess what? Can’t do anything about it. We can have discussions about securing our borders, but there’s nothing we can do about it.”National Democratic organizations that do not typically involve themselves in local elections are using abortion policy to promote and raise money for candidates who back abortion rights.Emily’s List, a group that backs women who support abortion rights, has endorsed mayoral candidates in Jacksonville, Fla., Madison, Wis., and Lincoln, Neb.In Lincoln, where Mr. Biden won 54 percent of the vote in 2020, Mayor Leirion Gaylor Baird, a Democrat, said her constituents had demanded to know what she could do about proposed legislation in the Nebraska Legislature that would restrict abortion rights. Her answer: speak out against the bills.Mayor Leirion Gaylor Baird of Lincoln, Neb., has urged constituents to speak out in favor of abortion rights. Madeline Cass for The New York TimesVoters, Ms. Gaylor Baird said, are “much more interested in knowing where people stand. So I expect that people will want to know where I stand on this issue, even if it isn’t a local issue typically.”Her main opponent, Suzanne Geist, a Republican state senator who has sponsored bills to restrict or ban abortion in Nebraska, said her actions in the State Capitol should have little bearing on how she would run the state’s capital city. She said she would prefer to focus on issues like public safety and the health of the city’s business community.Talking about abortion, Ms. Geist said, is “a way of avoiding what the present issues are and trying to get the public wrapped around something that really has nothing to do with the mayor’s office or the mayor’s race.”Suzanne Geist, a Republican state senator running for mayor of Lincoln, Neb., said that talking about abortion was “a way of avoiding what the present issues are and trying to get the public wrapped around something that really has nothing to do with the mayor’s office or the mayor’s race.”Madeline Cass for The New York TimesPast opposition to some abortion rights has become a political liability even for candidates who support them now. In Chicago, Paul Vallas, the former Chicago Public Schools chief executive who is running for mayor, is being attacked by his more liberal opponent, Brandon Johnson, for a 2009 television interview in which Mr. Vallas said, “Fundamentally, I oppose abortion.”Mr. Vallas’s statement, which he made when he being asked about possibly running for state office as a Republican, came after he had declared himself “personally pro-choice” but said he would favor banning some late-term abortions.Mr. Johnson is now broadcasting ads with a clip of Mr. Vallas’s statement that he opposed abortion; Mr. Vallas has responded with advertising declaring that he supports abortion rights.In an interview on Sunday at a Greek restaurant, Mr. Vallas said Mr. Johnson had taken his past abortion comments out of context.“It’s had some impact,” he acknowledged.In other races, municipal candidates are trying to find ways to make their cities have some influence over abortion access.Daniela Velázquez, a public relations executive running for the St. Louis Board of Aldermen, has proposed providing money for women seeking abortions to travel across the Mississippi River to Illinois, where the procedure remains legal. While abortion became illegal in Missouri after the Supreme Court’s decision, Ms. Velázquez said many in St. Louis supported abortion rights.“I have been knocking on doors and people have looked at our lit and been like, ‘Oh, you know, pro-choice,’” she said. “Then they say, ‘Yeah, I’m going to vote for you.’”Democrats are open in their belief at the current moment, the best way to win votes is to focus on the abortion fight.“Abortion and reproductive rights is the No. 1 issue in 2023,” said Ben Wikler, the chairman of the Democratic Party of Wisconsin, which has given hundreds of thousands of dollars to back Mr. Genrich in Green Bay and Mayor Cory Mason in Racine, who is making similar arguments there. “It’s the No. 1 issue that moves voters that normally vote Republican to vote for someone else and it’s the No. 1 issue to get Democrats off the couch and casting ballots.”Beyond the Green Bay mayoral election, abortion is a major issue in Wisconsin’s race for the State Supreme Court, which will finish on Tuesday and is likely to decide whether the procedure remains illegal in the state. Kayla Wolf for The New York TimesIn November, Racine asked voters on the midterm-election ballot if Wisconsin’s 1849 abortion ban should be repealed — and 71 percent said yes. Mr. Mason is now running television ads highlighting his stance in favor of abortion rights and attacking his opponent.Abortion, Mr. Mason said, comes up in his discussions with voters as much as snow plowing, public safety and housing.“These two big issues around freedom, the freedom to vote and the freedom to make your own health care decisions, they are every bit as front and center in this race as anything else that we deal with at the municipal level,” Mr. Mason said.Mr. Mason’s opponent, Henry Perez, a Republican city alderman opposed to abortion rights, said voters in Racine did not care much about the issue. He said that he did not remember how he had voted in the November abortion referendum, and that too much fuss was being made over abortion being banned in Racine when it was available across the state line in Illinois, roughly 25 miles south of the city.“A lot of people I’ve talked to say, ‘Henry, abortion, really?’” Mr. Perez said. “What do we care about it here? I mean, it’s not a thing that we do. And there’s always options like going out of town, you know, or going over to the next state to take care of an abortion if they need to.”Mitch Smith More