In the aftermath of the deadly mass shootings in El Paso and Dayton, Ohio, President Trump on Monday proposed no fewer than a half-dozen ideas to reduce gun violence — a mishmash of proposals that varied from the legislatively possible to the ill-defined and implausible. 

Trump decried “gruesome and grisly video games” and urged the public to begin lessening their influence, even though researchers say there is little evidence that ties video games to violent behavior. He promoted a bipartisan effort led by Sen. Lindsey O. Graham (R-S.C.), one of his closest congressional allies, to help confiscate firearms from those deemed unfit to possess them.

Those statements came after Trump, in a pair of tweets, called for strengthening background checks for gun purchases but in tandem with “desperately needed immigration reform” — tying the two contentious issues together for reasons that neither he nor White House officials chose to explain.

The end result was a muddled response that spurred momentum on some fronts — such as promoting extreme risk protection orders commonly known as “red-flag” laws — but prompted backlash on others. There was no sign that Congress would return to Washington to address the issue.

“While he is, in some ways, talking about gun violence, he continues to conflate gun violence to other things,” said Kyleanne Hunter, vice president of programs at Brady, which advocates for gun restrictions. “Almost every country has a mental-health problem. Every country has video games. Every country has immigrants and migrants and refugees.

Hunter added: “America is alone in the fact that they have a gun violence problem.”

White House media officials did not respond to questions asking for specifics behind what Trump had proposed in a 10-minute address earlier Monday, such as how to reduce the influence of video games or regulate the Internet to prevent the radicalization of what Trump called “disturbed minds.” 

Nor did they elaborate on why Trump linked the issue of background checks to immigration, since neither shooter in Dayton nor El Paso was an immigrant. Law enforcement officials are also investigating a manifesto filled with anti-immigrant views and warning of a “Hispanic invasion” that the suspect in the El Paso massacre, Patrick Crusius, purportedly wrote. 

Republicans offered a more charitable explanation: that Trump was tossing out several ideas as a starting point for discussion.

“The most optimistic view of it is, he’s putting everything on the table,” said Rep. Peter T. King (R-N.Y.) on Monday. “He’s the ultimate negotiator and bargainer and he realizes that if you’re going to go take a different position than you’ve taken before, then you have to offset it somehow.” 

Though discussions are preliminary, several senior Republican officials on Monday pointed to extreme risk protection orders as the most plausible legislative route, particularly if Trump continues to endorse it and give GOP lawmakers political cover to back the most anodyne of gun restrictions.

“A lot of this will be driven,” one senior Republican aide said, “by how active Trump is.” 

Graham and Sen. Richard Blumenthal (D-Conn.) are writing legislation that would offer federal grants and other incentives for states to develop laws implementing emergency risk protection orders. Those statutes would allow family members, law enforcement officials and others to petition a judge to bar firearms from someone they believe is an imminent threat to themselves or others. Seventeen states and the District already have such laws on the books, according to the Giffords gun-control group.

Trump has been known to endorse gun restrictions and subsequently backtrack amid opposition from conservatives, and there were some early signs of consternation as some GOP officials noted Graham, who leads the Senate Judiciary Committee, didn’t consult other Republicans on the panel on his plans. 

But Graham said he spoke with Trump earlier Monday and that the president seemed supportive. 

In an interview, Blumenthal said his goal was to introduce the bill by October or earlier. 

“The president’s support would be a breakthrough if he is about action as well as words,” Blumenthal said. But “this statute is one of only a number of measures that are necessary.”

Hunter, the Brady official, said suspects in both El Paso and Dayton would have been candidates for protective orders, which she said have been “incredibly effective” at preventing gun violence.

Senate Majority Leader Mitch McConnell (R-Ky.) said Monday that he had spoken with Graham, Sen. Roger Wicker (R-Miss.) and Sen. Lamar Alexander (R-Tenn.), who chair the relevant committees that would develop policy mentioned by Trump in his address. McConnell, who has been under pressure from Democrats to take up a universal background checks bill that passed the House earlier this year, stressed that “Senate Republicans are prepared to do our part.” 

“Only serious, bipartisan, bicameral efforts will enable us to continue this important work and produce further legislation that can pass the Senate, pass the House, and earn the president’s signature,” McConnell said in a statement that never mentioned the word gun.

That would almost certainly exclude expanded background checks — an effort that faltered in the Senate in the aftermath of the mass shooting at Sandy Hook Elementary School in Newtown, Conn., that led to the deaths of 20 children and six school employees.

Sens. Joe Manchin III (D-W.Va.) and Sen. Patrick J. Toomey (R-Pa.), who wrote legislation that would expand background checks to nearly all firearm sales, each spoke separately with Trump and came away believing that the president is willing to work on strengthening background checks. 

But Toomey also acknowledged that if the legislation came up in the Senate, it would not pass. It failed to advance in a Democratic-led Senate, with only four GOP senators supporting the legislation in 2013. Of them, only Toomey and Sen. Susan Collins (R-Maine) remain in office. 

“This isn’t going to happen tomorrow and if we force a vote tomorrow, then I think the vote probably fails,” Toomey said. “So if you want a successful outcome, which is what I want, then I think you work towards developing the coalition and the consensus so that you actually get the right outcome.”

Other parts of Trump’s address were far less well received, particularly by experts. 

In particular, Trump’s mention of mental health in his address drew criticism, including his reference that “mental illness and hatred pulls the trigger, not guns.” Most studies on mental illness and shootings have found only a fraction of mass shooters have diagnosed mental health issues. 

“These over-simplistic arguments that mental health or video games are causing shootings aren’t going to get us to the solution,” said Arthur C. Evans Jr., a psychologist and the CEO of the American Psychological Association. “My most generous way of thinking about it is, it’s hard for the average person to look at horrific events and not conclude only someone who’s out of their mind would do that . . . a less generous way of thinking about it is that it’s a politically expedient way to avoid talking about gun control or safety laws around guns.”

Trump has also moved to loosen restrictions on the mentally ill from buying guns. Early on in his administration, the Republican-controlled Congress passed legislation overturning an Obama-era regulation that barred certain people with mental-health issues from purchasing firearms. 

The president on Monday also said he was directing the Justice Department to draft a proposal that would hasten the death penalty for those convicted of committing certain federal hate crimes and mass murders.

While the Justice Department can already seek the death penalty for federal firearms offenses in which someone has been murdered, the agency is focusing on proposals that, in cases of mass shootings, would expedite the timeline for suspects to face the death penalty, according to officials.

The administration only recently announced it would seek to resume executions of federal prisoners who had been awaiting capital punishment last month after a 16-year moratorium on the death penalty.

William Wan, Paul Kane, Felicia Sonmez and Matt Zapotosky contributed to this report.