Battles over union fees are also emerging in other states. Republican legislators in Missouri and New Mexico are weighing similar measures. In Kentucky, where a split Legislature and a Democratic governor pose obstacles to a statewide bill, leaders in more than a dozen counties have approved or are weighing measures, officials there said on Saturday, and efforts in six other counties are awaiting final approval.

And in Illinois, a long-held Democratic territory with Democratic supermajorities in the legislature, the new Republican governor, Bruce Rauner, announced an executive order barring state workers who opt out of unions from being forced to pay fees based on a constitutional argument, offering a new model for states where split partisan politics have slowed right-to-work policies.

Federal law already permits workers not to join unions. But these laws go further, permitting workers to not pay fees to them. Unions argue that the fees are fair for nonunion members who still benefit from the contracts they negotiate, and that without a requirement, their membership, financial support and very existence are threatened.

The effects of such measures are fiercely debated, with dueling experts and research papers.

In Michigan, the percentage of workers in unions has dropped to 14.5 percent from 16.6 percent before the changes. Yet in Indiana, the percentage of union members grew to 10.7 percent from 9.1 percent in 2012, a statistic some labor experts say shows how difficult it is to gauge the effects of such measures given other factors at play.

In Wisconsin, the percentage of workers in unions dropped to 11.7 percent in 2014 from 14.2 percent in 2010, before Mr. Walker took office.

Central to the new momentum behind the laws were sweeping Republican victories in state elections in 2010, when the party got full control — in the chambers and the governor’s office — of states that included Wisconsin, Michigan and Indiana. Republicans made more gains in 2014, now controlling 68 of the 98 chambers around the country and the most state legislative seats since 1920. But it was the victories in 2010 that set off a new flood of right-to-work legislation in the Midwest, which had rarely seen it.