President Donald Trump said investigators were making progress in tracking down the gunman who fired a single rifle shot on Wednesday that struck the neck of Kirk, a 31-year-old author and podcast host who helped galvanise the conservative youth vote and return Trump to the White House.
Officials were still calling the man captured on video a person of interest, not a suspect, but placed him at the scene of the crime at Utah Valley University in Orem, Utah.
Kirk was on campus for one of his student-outreach events, featuring his trademark format of taking questions and challenging opponents to debate on the most polarising issues of the day, including gun violence and race. About 3000 people were in attendance.
The shooting has punctuated the most sustained period of US political violence since the 1970s.
Reuters has documented more than 300 cases of politically motivated acts of violence across the ideological spectrum since supporters of Trump attacked the US Capitol on January 6, 2021.
Trump himself has survived two attempts on his life in 2024, one that left him with a grazed ear during a campaign event and another two months later foiled by federal agents.
Video played at a media briefing late on Thursday showed a man walking across the roof of the building where the gunfire originated, before climbing down and dropping to the ground and leaving the campus.
Across the road, he entered a small wooded area where officials recovered what they described as a high-powered, bolt-action rifle they believe was used in the shooting.
Bolt-action rifles, unlike self-loading semi-automatic rifles often used in mass shootings, are popular with American game hunters, target shooters and snipers in militaries around the world.
They require the manual loading of each cartridge into the chamber with a turn of the bolt, but are perceived as more accurate at longer ranges when a single, fatal shot is all that is needed.
Utah Department of Public Safety Commissioner Beau Mason said the man left some palm impressions and smudges where investigators were looking to collect DNA.
"There's a shoe imprint where we believe the suspect is clearly identified as wearing Converse tennis shoes," Mason said.
Utah Governor Spencer Cox, appearing at a media briefing with FBI Director Kash Patel and other officials, asked for the public's help in identifying the slender young man, whose appearance was partially concealed by a dark baseball cap and sunglasses.
"We cannot do our job without the public's help right now," Cox said.
"So far, we've received more than 7000 leads and tips. I would just note that the FBI hasn't received this many digital media tips from the public since the Boston Marathon bombing" of 2013.
The FBI offered a $US100,000 ($A150,000) reward for information leading to the arrest of Kirk's killer.
Lawmakers, commentators and online sleuths have already filled social media and message boards with speculation about the killer's identity and blame-casting about his ideology.
Kirk, 31, a husband and father of two, was dear to many in Trump's MAGA political movement.
Vice-President JD Vance credited him with helping Trump win the 2024 presidential election and select people appointed to the Trump administration.
Vance cancelled a trip to New York and instead travelled to Utah to see Kirk's family and to fly them and Kirk's casket home to Arizona aboard Air Force Two.
Trump said he would award Kirk the Presidential Medal of Freedom, the nation's highest civilian honour.
"Charlie Kirk was a great person, a great man - great in every way, especially with youth" Trump told reporters.
"Investigators are making great strides on finding Charlie Kirk's assassin. Hopefully we'll have him and we will deal with him very appropriately," Trump said.
