Governors may not have a vote on whether to impeach the president, but those who would be governor have different opinions on how much that should matter to voters.

In the three Southern states that will choose a chief executive next month, the Republican candidates strongly oppose congressional Democrats’ efforts to impeach President Trump and are trying to pin down their Democratic opponents on the issue. The Democrats running in the states, which Trump carried in 2016, would prefer to talk about something else.

Consequently, while impeachment won’t be part of the gubernatorial duties, it is part of the gubernatorial debate.

On Monday, Kentucky GOP Gov. Matt Bevin again raised the issue with his Democratic opponent, Democratic Attorney General Andy Beshear. On social media the Bevin campaign released a video, “renewing calls for Absent Andy Beshear to be honest with Kentuckians and state once and for all whether he supports impeaching and removing President Trump from office.”

“As 95% of congressional liberals vow to impeach, Andy refuses to answer the simple question of whether he agrees with his own party that President Trump must be removed from office,” the Bevin campaign said.

Mr. Bevin this month at the capital railed against media he considers supportive of the Democratic campaign, saying the press has allowed Mr. Beshear to skate on the biggest political topic of the day.

Mr. Trump, who won Kentucky in 2016 with 62.5% of the vote, remains popular in the Bluegrass State.

When asked where Mr. Beshear stood on the question, his campaign referred The Washington Times to a statement it released after Mr. Bevin’s Frankfort press conference

“As Andy Beshear has already said publicly numerous times, he is the state’s top prosecutor and relies on evidence and fact,” it reads. “Just like many other Kentuckians, the only information he has right now is what he’s seen in different news reports. He believes that if Congress moves forward, any proceedings should be nonpartisan and focus on facts and evidence.”

Mr. Beshear’s move is smart politics, experts said.

“Beshear is probably wise to deflect questions on impeachment,” said Scott Lasley, a political science professor at Western Kentucky University. “The less he can say on the issue the better. Bevin is clearly emphasizing his close relationship with Trump.”

Republicans counter that the Democrats are dodging the biggest political issue, and the Republican Governors Association has tried to connect the dots between outspoken anti-Trump politicians who have pushed for impeachment and the Democratic campaigns.

In Mississippi, Republican Lt. Gov. Tate Reeves is a staunch supporter of Mr. Trump and has noted that the Democratic candidate, Attorney General Jim Hood, has ties to groups driving the impeachment narrative, such as California Attorney General Xavier Beccera, who has made something of a cottage industry suing the Trump administration.

“Hood has taken huge donations from resistance leaders — the very people pushing this impeachment,” campaign spokesman Parker Briden told The Washington Times. “Tate has been firm that this is a sham, and Mississippi deserves a leader who won’t back down to the National Democrats. Jim Hood has proven that’s all he’ll ever do.”

That may strike some as thin connective tissue, yet there is no question Mr. Hood has tried to distance himself from his Democratic Party’s lurch toward impeachment.

“Hood is not touching impeachment,” said Douglas Bristol, a history professor at the University of Southern Mississippi who based his conclusion on discussions with Hood supporters at the college’s homecoming. “He wants to win Republican support, so there is little reason to bash Trump.”

In a state Mr. Trump carried with 57.8% of the vote, Mr. Hood presents himself as a pickup-driving hunter. The Hood campaign has issued a blanket statement when asked about impeachment that offers little clue about how Mr. Hood would vote or thinks the Mississippi congressional delegation should vote.

“All the craziness and gridlock in Washington has created a positive response here in Mississippi, where moderates in both parties are coming together to move our state forward,” the statement reads. “Seeing Republicans, Democrats and independents support our campaign lifts my spirits that we can create an economy where our kids don’t have to leave the state.”

Mr. Trump won Louisiana three years ago with 58% of the vote and remains popular in the Pelican State, a position that has not gone unnoticed by incumbent Democratic Gov. John Bel Edwards.

While tapes and stories have dripped out about his personal dislike of the president, in public Mr. Edwards has touted his solid working relationship with the Trump administration and indicated he would maintain that in his second term.

The Edwards campaign did not reply to questions about impeachment or where he thinks the Louisiana congressional delegation should stand on the issue. Only the lone Democrat in the delegation, New Orleans-based Cedric Richmond, has backed the push.

“I think he’ll continue to parse words like he has been doing,” said John Couvillon, a Louisiana pollster and political consultant who has followed the governor’s race closely.

Although Mr. Edwards is unlikely to peel off any more conservative support than he already enjoys, it simply makes political sense for him to avoid the subject, Mr. Couvillon said.

“He still has to be more nuanced,” he said. “He doesn’t want to exacerbate the situation by looking hostile to the president.”

That neutrality could prove harder for Mr. Edwards to maintain in the runoff, as the top Republican challenger in Saturday’s primary election, Baton Rouge businessman Eddie Rispone, has made his backing of Mr. Trump a cornerstone of his rookie pitch.

His gubernatorial bid marks his campaign debut, and he wound up in the runoff against Mr. Edwards partly because he poured more than $11.5 million of his own money into the race.

The Rispone campaign did not respond to requests for comment on the impeachment topic, but Mr. Rispone stood on stage with President Trump in Lake Charles last week as the president in explicit language labeled the impeachment effort “BS” and called it “one of the greatest con jobs in U.S. history.”

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