LOUISVILLE, Ky. — The next battle for gun rights activists is over expanding the right to carry concealed weapons, with the NRA planning to fight for national reciprocity so holders in one state can travel to others without having to stow their guns.

Chris Cox, who heads the National Rifle Association’s Institute for Legislative Action (NRA-ILA), said that while all 50 states have some form of right-to-carry, the goal is federal reciprocity.

“We need a Congress to pass national right-to-carry reciprocity and a president to sign it into law, and we’re not going to stop until we get it,” Mr. Cox said at the NRA’s annual membership meeting this weekend.

Milwaukee County, Wisconsin, Sheriff David Clarke also called for a coordinated effort to persuade Congress to pass right-to-carry reciprocity legislation.

“We need to band together with the NRA and every gun-supporting lobby in America and pressure this GOP-controlled Congress to make it priority legislation to pass federal concealed carry license reciprocity in the United States of America,” Sheriff Clarke said.

Presumptive Republican presidential nominee Donald Trump said in his position statement on the Second Amendment that concealed carry permits should be valid in all 50 states. He touted his own status as a permit holder as he fired up the grass roots in Louisville.

“There are 13 million right-to-carry permit holders in the United States,” he said. “I happen to be one of them. … Nobody knows that. Boy, would I surprise somebody if they hit Trump.”

He acknowledged his large contingent of Secret Service agents but said those who carry concealed weapons are statistically among the most law-abiding in the country, “so they have the right to carry.”

Sen. John Cornyn of Texas, the Senate’s second-ranking Republican, has introduced legislation to allow concealed carry permit holders to carry in any state that also allows concealed carry. Several House Republicans have introduced similar bills.

Charlie Curie, 63, of Ohio, said federal right-to-carry reciprocity is one of the most important pieces of legislation that Republicans on Capitol Hill could pass. He said a Republican-controlled Congress could pass such a bill but would need a different president to sign it into law.

“To make that law more uniform would be very, very beneficial not only to the concealed carry permit holders, but also the law enforcement. … I think it’s a win-win on both sides,” he said.

Doug Tucker, 59, of Kentucky, said it’s wrong that people could drive through states such as New York and New Jersey with guns in their cars “and not know their stupid laws and be charged as a felon because you have a gun in your state [that’s] not registered in their state.”

The House passed concealed carry reciprocity legislation in 2011, though the bill didn’t go anywhere in the Senate, which was controlled by Democrats at the time.

Still, the issue isn’t cut clearly along partisan lines.

A handful of House Republicans opposed the legislation because they feared loosening gun laws, and some conservatives with strong gun rights records said it trampled another conservative principle: states’ rights.

While the NRA-ILA is focused on reciprocity, Mr. Trump and other prominent Republicans who spoke at the convention said they want legislation to repeal gun-free zones.

“We’re getting rid of gun-free zones, OK? I can tell you that,” Mr. Trump said. “We’re getting rid of them.”

Mr. Trump’s comments about gun-free zones drew a swift rebuke from likely Democratic presidential nominee Hillary Clinton, who said Mr. Trump wants to allow every school in the nation to have guns in classrooms.

“Parents, teachers and schools should have the right to keep guns out of classrooms, just like Donald Trump does at many of his hotels, by the way,” Mrs. Clinton said at an event in Florida. “This is someone running to be president of the United States of America, a country facing a gun violence epidemic, and he’s talking about more guns in our schools.”

Other critics noted that Mr. Trump was delivering his speech in a gun-free zone. Though people were allowed to carry their firearms into the Kentucky Exposition Center, they couldn’t bring them into the convention because of rules set by the Secret Service.

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