FOX CROSSING, Wis. — Gov. Scott Walker made a national name for himself in the Republican Party by cutting the power of public sector unions eight years ago, only weeks after he swept into office in Wisconsin. On Friday, he signed legislation to cut the power of the Democrat who defeated him, a final act of a tumultuous tenure that moved Wisconsin firmly to the right.

The response from Democrats was swift and furious.

Tony Evers, the Democrat who beat Mr. Walker in a hard-fought election last month, said the departing governor “chose to ignore and override the will of the people.” Liberal groups, including one led by Eric H. Holder Jr., a former United States attorney general, pledged to sue. Senator Tammy Baldwin of Wisconsin sent her Democratic supporters an email calling the move “a craven partisan attack on democracy” and soliciting donations to her “Vote ’Em Out Fund.”

The tactic by Mr. Walker and his allies came as part of an increasingly fractious struggle over power in the states, following a model set in North Carolina, where Republicans in 2016 tried to restrict the power of the governor after a Democrat was elected.

Similar scenarios were playing out elsewhere. In New Jersey, Democrats were seeking to make Republicans a permanent minority by, in essence, writing gerrymandering into the State Constitution. In Michigan, Republicans this week were contemplating limits on incoming Democrats, and the outgoing governor, Rick Snyder, on Friday signed bills scaling back a minimum-wage increase and a paid-sick-leave measure that had been slated for statewide votes until Republicans intervened.