In voting elsewhere Tuesday, Gretchen Whitmer, a former Democratic leader in the Michigan State Senate, claimed her party’s nomination for governor, setting up a crucial test for Midwestern Democrats and organized labor in November. Ms. Whitmer will face Bill Schuette, the state attorney general. Republicans have dominated Michigan for most of the last decade and Mr. Trump carried the state narrowly in 2016. The governorship is a vital prize for Democrats seeking a comeback there.

In the Senate race in Michigan, John James, an African-American Republican who also had Mr. Trump’s backing, won the nomination to challenge the Democratic incumbent, Debbie Stabenow. And in a race to replace longtime representative John Conyers Jr., who resigned in December after allegations of sexual misconduct, Rashida Tlaib took a major step toward becoming the first Muslim woman in Congress by winning the primary to compete in November in a safely Democratic seat.

In Missouri, Josh Hawley, the Republican state attorney general, will face off against Senator Claire McCaskill, a vulnerable Democrat.

And while Mr. Trump crowed about lifting Mr. Balderson in Ohio, his ability to propel a much-closer ally in the Republican primary for Kansas governor appeared uncertain Wednesday morning. One day after he received an endorsement from Mr. Trump that he had aggressively lobbied for, Secretary of State Kris Kobach was clinging to a very slim lead over Gov. Jeff Colyer with almost all precincts reporting. However, Mr. Colyer was enjoying an advantage in Kansas’s most populous jurisdiction, Johnson County, where nearly all the remaining votes were to be counted.

If Mr. Kobach, who has made a national name for himself for devising crackdowns on immigration and voting rights, were to lose, it would snap Mr. Trump’s streak of success in Republican primaries. The president has crowned Republican nominees from Georgia to South Carolina to Alabama in recent nominating contests. In the case of Kansas, though, Mr. Trump defied White House aides and other high-level Republican officials who urged him to stay out of a race involving a sitting governor. Many Republican leaders, and most all Democrats, believe Mr. Kobach would be a weaker general election candidate for the G.O.P. and perhaps imperil their grip on the state’s governorship

But it was the Ohio congressional race that transfixed Democrats and Republicans as one of the marquee midterm battles of the summer. The Republican Party’s tactics that helped Mr. Balderson are now likely to be reproduced across the country this fall, as Republicans defend their congressional majority in similar districts stocked with the right-of-center suburbanites who proved so elusive in the special election.

Republicans prevailed in Ohio by wielding a set of divisive issues — including gun rights, immigration restriction and taxes — that mobilized just enough voters on their side, particularly in the exurban and more rural reaches of the district. And they invoked Ms. Pelosi at every opportunity.