With the Democrats’ sit-in on the House floor, gun-control debates among lawmakers may have reached new heights of polarization. But on at least one of the proposed policies, advocacy groups typically at odds with one another are finding themselves on common ground.

The proposals in question would bar gun sales to those appearing on a “no-fly list” – a secret list of people prohibited from flying on airplanes in the United States. Twice this past week, the American Civil Liberties Union sent letters to the Senate urging lawmakers to vote against three bills using no-fly lists as a basis for gun restrictions. The group cites “the lack of adequate due process safeguards” as a chief basis for its opposition.

The National Rifle Association also has raised due-process concerns about the bills.

“We’re not working together with the NRA,” says ACLU deputy director Chris Landers, “but we’re in some places making similar arguments.”

“An important place we differ is that the ACLU does believe that there are ways that a government can regulate guns that are not going to raise due-process or equal-protection problems,” said Mr. Landers. “Relying on watch lists, though, is not one of those ways.”

Guns-rights advocates are staking their opposition on similar grounds.

“This is an issue where it’s not just law-abiding Americans being denied their Second Amendment rights, it’s Fifth Amendment [due process] rights, too,” says NRA public affairs director Jennifer Baker.

The group doesn't take a position on no-fly lists more generally. “We’re a single-issue organization,” says Ms. Baker, “so our mission is to protect and defend the Second Amendment.”

The right to bear arms, she added, “is enshrined in the Constitution. Flying on a plane is not the same as a Second Amendment right.”

Nor does the NRA oppose every gun-control bill that uses the no-fly list. It backed a Republican-sponsored version that would have notified the US attorney general’s office any time someone on a no-fly list – or a much larger terrorism watch database – tried to buy a gun. But to block the sale, the office would be required to respond within 72 hours of the notification, with proof that the potential buyer has ties to terrorism.

Other conservative groups, think tanks, and gun-rights advocates have taken exception to no-fly list gun laws. And in an interview on CNN on Thursday morning, House Speaker Paul Ryan said Republicans in the chamber refused to consider new legislation that would “take away a citizens’ due process rights.”

Even after deploying an unusual series of tactics in the House this week, including a failed last-minute attempt to tack an amendment on an annual appropriations bill, Democrats weren’t able to wrangle a vote on new gun-control measures. On Thursday afternoon, they announced they would bring an end to their sit-in, with Republican leaders having declared the chamber in recess.

Rep. John Lewis (D) of Georgia, the civil rights era icon who led this week's sit-in for gun-control, ended up on an airport screening list in 2004, when he was reportedly stopped 35 to 40 times in the space of a single year. His office did not respond to requests for comment.

In 2014, Representative Lewis also joined 20 House Democrats in signing a letter to the Department of Homeland Security expressing concern that the department was still not providing “effective means of redress for unfair or incorrect designations” on federal watch lists.

On Thursday, Lewis urged supporters to keep up momentum on gun control when legislators returned on July 5.

"We are going to win," he said. "The fight is not over.”