WASHINGTON — Senate Republican leader Mitch McConnell dispatched his tea party challenger with ease Tuesday night, and Democrats turned to two women, Alison Lundergan Grimes to oppose him in Kentucky and Michelle Nunn to fight for Georgia, in elections next fall with control of the Senate at stake.

Setting up a third high-profile race, Democratic Sen. Mark Pryor of Arkansas and his Republican challenger, Rep. Tom Cotton, were unopposed for their parties’ nominations.

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With his wife, Elaine Chao, looking on, Kentucky Sen. Mitch McConnell addresses his supporters following his victory in the Republican primary Tuesday at the Marriott Louisville East in Louisville, Ky. The Associated Press/Timothy D. Easley

On the busiest primary night of the year to date, Democrats eyeing a return to power in the Pennsylvania state Capitol nominated businessman Tom Wolf to oppose Republican Gov. Tom Corbett’s bid for a second term.

Republican primary struggles between establishment-backed conservatives and tea party-favored rivals were a dominant feature of the evening, as they had been earlier in North Carolina and will be later in Mississippi, Kansas and Alaska. Republicans must gain six seats to win a Senate majority, and party leaders have made it a priority to avoid the presence of candidates on the ballot this fall who are seen as too conservative or unsteady — or both — to prevail in winnable races.

McConnell, a five-term lawmaker and the embodiment of the GOP establishment, was pulling 60 percent of the vote in Kentucky. Challenger Matt Bevin was gaining 36 percent.

For Democrats, Tuesday night was a chance to showcase challengers — both of them women — in the rare states where the party has hopes of picking up GOP-held seats.

Grimes, a prize Democratic recruit, was piling up 76 percent in a four-way race, winning her Kentucky primary with ease.

She and McConnell wasted no time turning their attention to the fall campaign.

“Make me the majority leader and Kentucky will lead America,” McConnell said in an appeal to home state pride, adding that he would use his power to check President Barack Obama’s agenda.

Grimes said Obama wasn’t on the ballot, and responded forcefully to some of the campaign barbs that have already come her way. “‘I am not an empty dress. I am not a rubber stamp. And I am not a cheerleader. I am a strong Kentucky woman,” she told cheering supporters in Lexington.

In Georgia, Nunn, whose father was a four-term Democratic senator from the state, easily outpaced her Democratic rivals and awaited the outcome of the GOP primary to learn her opponent for the fall.

Seven Republicans vied for that nomination, and a two-way July 22 runoff appeared a certainty. With votes counted in 45 percent of the precincts, businessman David Perdue led with 30 percent of the vote, Rep. Jack Kingston had 29 percent and former Secretary of State Susan Handel was running third with 19 percent.

Along with Perdue, Kingston and Handel, Reps. Phil Gingrey and Paul Broun also were on the Georgia Republican ballot, and the presence of three incumbent lawmakers in the Senate race assured a large turnover in the state’s House delegation come January.

Some Republican primary voters said they had made up their minds based on more than the names on the ballot.

“I’m conservative, but I think most of the tea party people are a little too extreme,” said David Reynolds, 63, of Union, Kentucky, after voting in his state’s Senate race. He said he cast his vote for McConnell over Bevin.

Bevin was backed by tea party groups in the state where they made their mark four years ago by sweeping GOP Sen. Rand Paul into office.

Outmaneuvered in 2010 when his preferred contender was defeated, McConnell responded this time by running ads featuring testimonials from Paul, and by hiring a top aide to Paul to run his own campaign.

For his part, Bevin stumbled through a campaign that included an appearance at a rally of cock-fighting supporters.

Plagued by low approval ratings, McConnell spent more than $9 million through the end of April on his primary campaign, according to Federal Election Commission figures. Bevin spent $3 million, and outside groups poured in $5 million more — a three-way deluge of television advertisements likely to continue through the fall.

The Georgia Senate race was fiercely expensive — $10 million had been spent on television commercials through the end of last week — and highlighted the divisions within the Republican party. Perdue relied on his background as a businessman, while Broun and Gingrey ran farther to the right. Handel sought to capitalize on the backing of former Republican vice presidential candidate Sarah Palin, and Kingston had the support of the U.S. Chamber of Commerce.

In Oregon, Republicans picking a nominee to oppose Sen. Jeff Merkley chose between state Rep. Jason Conger and Monica Wehby, a physician.

There were gubernatorial primaries five states.

In Georgia, Republican Gov. Nathan Deal defeated two primary challengers. State Sen. Jason Carter, grandson of the 39th president, was unopposed for the Democratic nomination.

Republican Corbett’s poor ratings in Pennsylvania drew a crowd in the Democratic gubernatorial primary. Wolf outpaced a Democratic field that included Rep. Allyson Schwartz, who began the campaign as the front-runner.

Arkansas primary voters set up a race between Republican Asa Hutchinson, a former congressman who also served in the Bush organization, and former Democratic Rep. Mike Ross.

A smattering of Republican House members faced primary foes, notably Rep. Mike Simpson of Idaho. Challenger Bryan Smith said the incumbent wasn’t conservative enough, and he drew early support from the Club for Growth in a bid to oust him.

Establishment groups rallied behind Simpson, and the Club for Growth quit running television ads for Smith weeks ago.

In Georgia, former Republican Rep. Bob Barr launched a comeback bid. Former Clinton administration official James Lee Witt ran unopposed in Arkansas for the Democratic nomination to the seat held by Cotton.