The outcome of Tuesday’s primaries will only provide more evidence for those who are ready to paint the 2010 cycle as the most anti-incumbent election in recent memory.

Despite his support from the Democratic establishment, Pennsylvania Sen. Arlen Specter’s 30-year Senate career came to an end with his defeat by sophomore Rep. Joe Sestak, who came from behind to win by a very comfortable 8-point margin Tuesday.

Further west, in Arkansas, Sen. Blanche Lincoln (D) was forced into a runoff with Lt. Gov. Bill Halter (D), whose surprisingly strong performance brought him within 2 points of Lincoln and means the Senator will have to sweat out another three weeks before the runoff.

And in Kentucky’s GOP Senate primary, Secretary of State Trey Grayson went down in spectacular fashion against ophthalmologist Rand Paul, who had been embraced by the tea party movement. Grayson was backed by the state’s party establishment, including Senate Minority Leader Mitch McConnell (Ky.), who helped recruit, fundraise and eventually cut a commercial for Grayson earlier this month. Paul’s victory, coming less than two weeks after tea party activists played a large role in defeating Utah Sen. Bob Bennett at the state GOP convention, is certain to energize the movement heading into the rest of the primary season, and Democrats hope that will spell trouble for other establishment candidates.

But before pundits get too caught up in the power of the anti-incumbent sentiment, it’s fair to ask: How much of what happened Tuesday was a result of the old adage that all politics is local?

Specter’s party switch at the beginning of the cycle was the key to Sestak’s victory. Specter’s three decades of service may have been less important in the minds of Democratic primary voters than the fact that he spent 28 of those years as a Republican. Sestak’s most effective ad was less about Specter’s length of service than it was about the political calculation that went into his party switch. The ad showed a video of Specter admitting that his change in party would allow him to be re-elected and ended with the tag line that “Arlen Specter switched parties to save one job. His own.”

In Arkansas, Halter’s strong showing has to be attributed in large part to the labor movement and other liberal groups, which decided to make an example of the moderate Senator this year. Those groups ran at least $4 million in TV ads and spent several million more on direct mail, radio and voter turnout efforts. Lincoln’s incumbency would likely not have been a problem for those labor groups if she had embraced the Employee Free Choice Act.

And in the Bluegrass State, Paul’s victory was certainly a milestone for the grass-roots tea party movement, and it’s undeniable that Grayson had the party establishment in his corner. But Paul entered the race with the support of a national fundraising network built by his father, Rep. Ron Paul (R-Texas), during his presidential campaign. That network allowed him to not be drowned out by Grayson, who early in the campaign appeared more intent on ignoring Paul — some insiders say out of fear of scaring him into making an Independent bid — than in defining him.

It’s hard to say that voters’ rejection of Grayson directly translates into a rejection of McConnell. A poll conducted earlier this week by Public Policy Polling, a Democratic firm based in North Carolina, found that McConnell remains popular among GOP primary voters. That survey found that 64 percent of primary voters believed the winner of the GOP primary should vote to back McConnell as the GOP leader in the Senate, while only 18 percent said no. Even among those who backed Paul, 58 percent were for keeping McConnell in his leadership post while only 22 percent said he should go. The poll had a 3-point margin of error.

Not to be overlooked with all the action on the Senate side Tuesday was the important victory by House Democrats in the special election to replace the late Rep. John Murtha (D) in Pennsylvania’s 12th district. Not only did the win extend Democrats’ streak of victories in competitive special elections to a half dozen dating back to the 2008 cycle, but it also served as a momentum-killer for the growing enthusiasm in Republican circles for retaking the House majority this cycle. The socially conservative, blue-collar district went narrowly for Sen. John McCain (R-Ariz.) in the 2008 presidential race and was seen as exactly the kind of battleground seat where Republican officials are looking to make major gains in the current political climate.

Democratic Congressional Campaign Committee Chairman Chris Van Hollen (Md.) on Tuesday night called the results a rejection of the GOP strategy of nationalizing the contest.

“For all of their bluster about building a national wave this year, including [Republican National Committee] Chairman Michael Steele’s guarantee of victory for Tim Burns, Republican policies were once again rejected when it came time to face the voters,” Van Hollen said in a statement.

In his statement Tuesday night, National Republican Congressional Committee Chairman Pete Sessions (Texas) said the special election loss was “undoubtedly disappointing,” but he said the committee would take the lessons learned and move forward.

“The bottom line is that the makeup of the House remains the same and our goal of winning back the majority in November has not changed,” Sessions said in a statement. “In just four days, we will have another election in another Democrat-held seat that stands to change the makeup of the House.”