Growing up in Bozeman, Mont., Dylan Heintz loved the picturesque views of the snow-capped mountains and the small-town charm. Things were cheap: His dad bought the family home for about $80,000.

These days, Bozeman feels less quaint. A steady stream of out-of-state transplants to Big Sky Country became a deluge during the pandemic, leading to soaring prices, a boom in luxury apartments that blot out the rustic scenery and a rash of higher-end businesses like Whole Foods. Drawn by Montana’s natural beauty and easy access to outdoor activities, the newcomers have created an affordability crisis and a local backlash that are transforming the state’s economy and politics.

“I love this place, but it’s just a tough place to live in,” said Mr. Heintz, 28, an auto body repairman. Rent has doubled in his trailer court, and he and his wife cannot afford to buy a home in town, leaving them considering a move to Florida. “There are a lot of out-of-staters that have some money, and they’re willing to pay above asking price. That definitely hurts people.”

The fresh population of wealthier residents — often retirees, technology workers able to do their jobs remotely and other big-city transplants — is one of the largest question marks hanging over Montana’s crucial race for Senate. As Jon Tester, the Democratic incumbent, looks to fend off Tim Sheehy, a businessman and retired Navy SEAL who is expected to capture the Republican nomination, tensions over the exploding growth will be a top issue in November.

And how the new Montanans vote could prove decisive.

On the surface, their presence might seem to benefit the embattled Mr. Tester, because a sizable chunk of them — 35 percent of arrivals in 2022 — hail from left-leaning states like California, Colorado, Oregon and Washington, according to census data analyzed by the real estate firm CBRE. Some political experts, though, believe the arrivals could tilt more to the right, noting a broader phenomenon in which conservatives have left their home states in part because of what they see as liberal overreach.