Asked which issues might help the president appeal across the divide, Mr. Wilmeth suggested national security and the economy. Yet he conceded the campaign’s reach on those issues has shrunk in Arizona as waves of newcomers have arrived who don’t share the conservative outlook: “We have Illinoisans and Californians who come here, and they’re bringing their values.”

Mr. Wilmeth said he sees more out-of-state license plates each day, and his perception is more than a hunch: The Grand Canyon State has been among the fastest-growing in the country in the past decade. Voters don’t report their state of origin at the polls, but it’s reasonable to assume these new arrivals have contributed to Republicans’ shrinking vote share, with margins of victory in presidential elections generally declining since 2004. Immigration plays a role as well, and an influx of Hispanic voters helped tilt neighboring Nevada and New Mexico toward Democrats years before Arizona became competitive. Yet voter registration is shifting Democratic even in heavily white places like Scottsdale, a top destination for incoming retirees and young professionals.

For the Trump campaign, the bright side of this trend may be that Arizona’s Republican old guard is highly aware of it—and is circling the wagons. “We need Trump to hold the line, 100%,” said James Hanson, 56, at Seamus McCaffrey’s Irish pub in downtown Phoenix. Conservative arguments on crime or abortion may not persuade the state’s growing liberal base, but Mr. Hanson is an example of how Democrats haven’t necessarily been any more successful in their efforts to recruit self-identified conservatives, even when their interests might seem to align with liberal causes.