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In a race considered key in Tuesday’s primaries, Kentucky’s U.S. Sen. Mitch McConnell, a Republican, handily beat his tea party challenger, Matt Bevin – another sign, many political experts say, that the most right-wing faction of the GOP is losing steam.

For so-called establishment Republicans, who hope to win six seats in the U.S. Senate, and gain control of that chamber, this year has been one of trying to beat back tea party candidates, viewed by the GOP as a hindrance to winning general elections in November.

On Tuesday, McConnell, who as a 30-year senator and party leader is about as establishment as they come, predicted that he and other mainstream Republicans would "crush" tea party candidates this year.

Bevin initially excited anti-establishment Republicans, but his campaign eventually collapsed under rookie mistakes and McConnell's overwhelming advantage in money, experience and organization.

One of the issues both Bevin and McConnell sparred over was immigration, with Bevin casting the senator as “pro-amnesty” and the senator at one point assailing Bevin, saying he’d insulted the people of Kentucky, when the candidate said that U.S. citizens did not want to do certain hard work that immigrants do.

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The Kentucky victory for establishment Republicans reflected the best evidence yet that Republicans are avoiding previous mistakes and improving their chances of controlling the Senate during President Barack Obama's final two years in office.

GOP voters again chose solidly conservative nominees while rejecting the most extreme and outlandish types who led the party to painful losses in 2010 and 2012.

While McConnell and Bevin criticized each other over immigration, they actually both harbor largely conservative views on the issue, supporting stricter enforcement.

In fact, the differences between tea party and non-tea party Republicans seem to be shrinking. Often it's merely tone and experience that separate them. Tone and experience matter, however, and Tuesday's GOP voters chose the less bombastic and unpredictable conservatives in most cases.

Kentucky Democrats, who are trying to guard against losing the majority in the Senate, are planning a tough offensive against McConnell with their candidate, Alison Lundergan Grimes.

McConnell is mindful of this, taking aim at her in his victory speech on Tuesday night, according to The Hill. The Senate Minority Leader called Grimes “a partisan’s partisan who’s been practicing party politics since she learned to talk.”

“Barack Obama’s candidates preach independence but they practice loyalty above all else,” McConnell said, according to The Hill. “I’m confident of this: Kentuckians will not be deceived. Alison Lundergan Grimes is Barack Obama’s candidate.”

Grimes has not missed a beat, meanwhile, responding that she is "not an empty dress...not a rubber stamp..not a cheerleader."

“Mitch McConnell would have you believe that President Obama is on Kentucky’s 2014 election ballot,” Grimes said. “Senator McConnell, this race is between you and me.”

Tea party candidates fared poorly in other Tuesday primaries, as well.

In Georgia, Republican voters rejected the two most outspoken tea party proponents, Reps. Paul Broun and Phil Gingrey. They set up a July 22 Senate runoff between two men who constantly emphasize their conservative credentials but leaven their rhetoric by wooing corporate support: Dollar General CEO David Perdue and Rep. Jack Kingston claimed the top two spots Tuesday and now begin a two-month runoff campaign.

Establishment Republicans once feared that Broun, who called embryology and evolution "lies straight from the pit of hell," would win the nomination and become the type of gaffe-prone, over-the-top candidate who killed great GOP Senate chances in Delaware, Indiana, Missouri and other states in 2010 and 2012.

In Oregon, Republicans chose pediatric neurosurgeon Monica Wehby, who supports abortion rights, to run against first-term Democratic Sen. Jeff Merkley this fall. Her opponents included state Rep. Jason Conger, who was endorsed by the Tea Party Nation and former presidential candidate Rick Santorum.

Arkansas' two uncontested Senate primaries officially set up a fierce November showdown between two-term Democratic Sen. Mark Pryor and freshman Republican Rep. Tom Cotton.

Tuesday's Republican elections continued earlier trends from states such as North Carolina. There, Republicans nominated state House Speaker Thom Tillis for the Senate, rejecting a tea party leader and a Baptist minister who were making their first runs for office.

North Carolina Democrats scoff at the notion that Tillis is "moderate," citing his self-described "conservative revolution" in the closely divided state. There and in many other states, Democrats will say the GOP Senate nominees are too conservative. They are happy to see Republican leaders play down the differences between themselves and tea party activists, as House Speaker John Boehner, R-Ohio, did Tuesday.

"Sometimes," Boehner told reporters, "there's not that big a difference between what you all call tea party and your average conservative Republican."

Before mainstream Republicans get too excited about Tuesday's Senate results, they might note that Democrats have solid, well-funded nominees waiting. Oregon's Merkley and Arkansas' Pryor — like North Carolina's Sen. Kay Hagan — have proven they can win statewide elections.

In Georgia and Kentucky, where Obama hardly campaigned in 2008 and 2012, Democrats are banking on two women with well-known political names. Michelle Nunn of Georgia is the daughter of former Sen. Sam Nunn. Kentucky Secretary of State Alison Lundergan Grimes is the daughter of a long-time Democratic Party leader.

Republicans need to gain six net Senate seats to control the chamber. Losing either McConnell's seat or the Georgia seat, which Saxby Chambliss is vacating, could kill those chances.

Nunn would rather have drawn Broun as her opponent. And Grimes would have been ecstatic to face Bevin rather than McConnell.

So far, Republican primary voters aren't handing those types of gifts to Democrats. That suggests they've learned the hard lessons of 2010 and 2012, when non-mainstream nominees lost winnable races.

Now the question is whether all Republican candidates — not just Constitution-quoting tea partyers — have moved too far right for moderate voters in November's general elections.

The Associated Press contributed to this report.

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