When he took the stage in downtown Eau Claire, Wis., on Tuesday night to rev up Democrats ahead of a critical State Supreme Court race, Gov. Tim Walz of Minnesota said he didn’t think name-calling would help things.

Then he called Elon Musk a “dipshit” and, later, a “South African nepo baby” with the power to cut government programs. The crowd roared.

Mr. Walz, his party’s nominee for vice president last year, is one of several Democrats who have referred to Mr. Musk’s immigrant background as they ramp up attacks on the billionaire’s powerful role in the Trump administration. At times, their language, casting Mr. Musk as a foreign outsider, has echoed aspects of President Trump’s own xenophobic insults of his political foes — although Mr. Trump’s remarks were typically directed toward elected officials of color, not white billionaires.

At a news conference last month, Representative Marcy Kaptur of Ohio said she wondered, “Which country is he loyal to? South Africa, Canada, or the United States?” Representative Nydia Velázquez of New York declared Mr. Musk should “go back to South Africa” at a recent protest. At a different protest, Representative Don Beyer of Virginia said, “We’re going to send Elon back to South Africa.”

Mr. Musk was born in South Africa in 1971, moved to Canada in 1989 and then to the United States during college. He obtained Canadian citizenship just before moving there and became a naturalized United States citizen in 2002, according to his biographer, Walter Isaacson.

Mr. Walz made his remarks at a town-hall-style event that coincided with the first day of early voting for a State Supreme Court race in Wisconsin. That election, which will be held on April 1, pits a liberal candidate against a conservative Trump ally who has drawn more than $13 million in backing from Mr. Musk. Wisconsin Democrats have seized on those contributions to cast Mr. Musk as the election’s main villain.