Gun control advocates are trying to make the issue a voting one in key U.S. Senate races across the country, hoping the subject will help take down Republican candidates in states like New Hampshire and Florida.

Former Rep. Gabrielle Giffords, who was severely wounded by a gunman in 2011, and her husband Mark Kelly are in the midst of a six-week, 14-state bus tour that includes stops in Florida, Ohio, Iowa and New Hampshire.

“In races up and down the ballot, 2016 is shaping up to be the year of gun safety,” said Kate Folmar, a spokeswoman for Everytown for Gun Safety, an advocacy group co-founded by former New York City Mayor Michael Bloomberg. “The political calculus has changed, and we are seeing candidates at all levels of government embrace gun safety as an issue to run — and win — on.”

Gun control advocates are closely eyeing U.S. Senate races in New Hampshire and Florida, where incumbent Republican Sens. Kelly Ayotte and Marco Rubio face tough re-election fights, and in Nevada, where Democrat Catherine Cortez Masto is trying to hold the seat being vacated by retiring Sen. Harry Reid.

But such single-issue groups also say they’re willing to support Republicans who are with them on the issue.

Mr. Bloomberg is supporting Sen. Pat Toomey, Pennsylvania Republican, through his Independence USA super PAC, and the political arm of Ms. Giffords’ gun control group, Americans for Responsible Solutions, endorsed both Mr. Toomey and Sen. Mark Kirk, Illinois Republican.

Ms. Giffords’ group, however, did take the unusual step of revoking its endorsement of Mr. Kirk after he made a derogatory remark about the heritage of his opponent, Democratic Rep. Tammy Duckworth, during a recent debate, for which he apologized.

“While we believe the path to responsible change and communities safer from gun violence must include Republican champions, we cannot continue to support Senator Kirk,” said Peter Ambler, executive director of Americans for Responsible Solutions PAC.

Activists associated with Everytown are supporting more than 20 Republican candidates across the country, even though the group has formally endorsed Hillary Clinton for president.

Ms. Giffords and Mr. Kelly are also supporting Mrs. Clinton and were scheduled to campaign with the Democratic nominee on Monday in Ohio.

“Certainly, Gabby and I are behind Secretary Clinton for a number of reasons — chiefly, it’s her support of common sense changes to our gun laws to keep communities safer from gun violence,” Mr. Kelly said on MSNBC on Monday.

The National Rifle Association, meanwhile, has been among GOP nominee Donald Trump’s most loyal backers, and has also formally endorsed Mr. Rubio, and Republican Sens. Richard Burr of North Carolina and Lisa Murkowski of Alaska.

The NRA has also attacked former Ohio Gov. Ted Strickland, a Democrat who is challenging incumbent Sen. Rob Portman, a Republican. The gun-rights group endorsed Mr. Strickland in 2010 but says he has since lost his way on the issue.

Both the presidential and U.S. Senate contests are also crucial for the future make-up of the U.S. Supreme Court, said Doug Deeken with the group Ohioans for Concealed Carry.

“That, I think, is really where gun owners look at this, is it’s not so much the presidency, it’s not so much the individual senator,” Mr. Deeken said. “It’s the legacy that outlasts every president — that of the Supreme Court.”

“How can we ensure that we get people who, when they read the Bill of Rights, don’t skip Amendment 2?” he said. “And that’s the big deal.”

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