(There is no evidence that the gunmen in the weekend massacres were influenced by video games or suffered from mental illness. And violent video games are common in many countries that do not experience mass shootings at anywhere near the level the United States does. The El Paso gunman’s manifesto railed against immigration and denounced the “Hispanic invasion of Texas.” )

Mr. Feinblatt said he had no interest in a deal that paired background checks with a Trump-approved immigration package.

“To make Trump’s narrow form of immigration reform a bargaining chip for gun safety legislation is just a cynical move to guarantee that nothing happens,” he said.

The N.R.A. left the door open on Monday to red-flag laws, also known as extreme-risk protection orders. “It has been the N.R.A.’s longstanding position that those who have been adjudicated as a danger to themselves or others should not have access to firearms and should be admitted for treatment,” the group said in a statement.

While the N.R.A. has long expressed openness to red-flag laws in principle, it has rejected most specific proposals as having insufficient due-process protections.

As the El Paso shooting was unfolding, more than 2,000 activists were at a hotel in Washington’s Woodley Park neighborhood for a training organized by Moms Demand Action for Gun Sense in America, an arm of Everytown.

On Saturday night, hundreds of them marched two miles to the White House — while Mr. Trump was crashing a wedding at his New Jersey golf resort — and then to the empty Capitol, chanting, “El Paso!” and “Not one more!”