That contrasts with his response to mass attacks like the deadly truck ramming in New York City just five days before the Texas shooting. After the truck attack on a bike path, he almost immediately reiterated his frequent demands for an overhaul of the immigration system and also called for the end of a specific visa lottery program.
It fits a pattern, according to one expert.
Such a focus is "ironic," Cohen added, because "many more casualties caused over the last several years in particular were through attacks by individuals who were born and raised in this country."
The statistics bear that out: Four of the five deadliest mass shootings in modern U.S. history involved American-born perpetrators, though one was the son of an Afghan immigrant. Seven of the 10 deadliest mass shootings were committed by white Americans.
But White House press secretary Sarah Sanders denied a day after the Manhattan truck ramming that Trump was trying to make political hay out of the attack. "The president has been talking about extreme vetting [of some immigrants] and the need for that for the purpose of protecting the citizens of this country since he was a candidate," she said at a press briefing.
Sanders added, "This isn't a new argument. This isn't a new position."
Trump has used a number of terms to describe different kinds of assailants.
In his Oct. 2 comments after the Las Vegas concert shooting, he referred to U.S.-born shooter Stephen Paddock simply as "a gunman" but in a tweet two days later called him "the demented shooter."
Trump was in Japan on Sunday when news broke of the church shooting in Sutherland Springs, and in his first public reaction he called suspect Devin Kelley "a very deranged individual."
Pointing to policy but only in some cases
In recent cases that involve perpetrators with foreign ties — including the 2015 San Bernardino, California, shooting; the Pulse nightclub shooting; and the truck attack in New York City — Trump has pivoted toward policy change suggestions within days or sometimes hours of the episode.
He drew a clear connection between certain shootings and the need for his then-proposed policies in one of his responses immediately after the Pulse nightclub shooting, in which Mateen killed 49 people.
"What has happened in Orlando is just the beginning. Our leadership is weak and ineffective. I called it and asked for the ban. Must be tough," Trump wrote in a tweet less than 14 hours after the first shots were fired.
"We've seen the attacks, from 9/11 to Boston to San Bernardino. Hundreds of individuals from other countries have been charged with terrorism-related offenses in the United States," he said in the speech.
He did not mention any mass shootings that occurred on U.S. soil, many of which were committed by people born in America.
Four of the five deadliest shootings in modern U.S. history were committed by men born in the United States: Stephen Paddock, who committed the Las Vegas shooting last month; Mateen, who was responsible for the Pulse nightclub shooting in Orlando; Adam Lanza, who opened fire at the Sandy Hook Elementary School in Newtown, Connecticut in 2012; and now Kelley in Texas.
In the fifth case, Seung-hui Cho, who was responsible for the 2007 Virginia Tech shooting, was born in South Korea but was a permanent U.S. resident at the time of the shooting.
In the 10 deadliest mass shootings in modern U.S. history, seven of the killers were white men, and three of the perpetrators were nonwhite.
Cohen, who has spent three years studying mass casualty attacks in the United States and Europe at Rutgers University in New Jersey, said the research is "finding a lot of behavioral and psychological characteristics that are common across this group of people, whether they're motivated by ideological objectives or something else."
"There is no religious, ethnic or socioeconomic profile — even among those motivated by extremist ideologies such as those of terrorist groups like ISIS," he said. "Regardless of motive, many of these offenders present common behavioral, psychological and life experience characteristics."
Cohen listed a number of such characteristics, including several that could apply to Kelley, such as a history of violent behavior, criminal activity and work and other life failures. It's unclear whether others — such as underlying mental or behavioral health issues, feelings of exclusion and perceived victimization — might apply in the latest mass shooting.