Kirk, a 31-year-old author, podcast host and close ally of US President Donald Trump, helped build the Republican Party's support among younger voters.
He was killed yesterday by a single gunshot as he gave a talk at a university in Utah in what Trump called a "heinous assassination."
FBI and state officials said the killer arrived on the campus a few minutes before the event began, a debate led by Kirk titled "Prove Me Wrong" outdoors in front of about 3,000 people at Utah Valley University in Orem, Utah, about 65 km south of Salt Lake City.
Security-camera videos show a person going up stairwells to get onto a roof before firing at Kirk, the officials told a press conference.
Kirk, a staunch defender of gun rights, was answering an audience question about mass shootings when the bullet struck his neck. Audience members fled in panic.
The shooter jumped off the roof and fled into an adjoining neighborhood, Robert Bohls, the FBI special agent in charge, told reporters.
Investigators found a "high-powered, bolt-action" rifle in a nearby wooded area, and were examining that along with palm prints and footprints for clues. On Thursday, with classes canceled, the roof of the building on the otherwise deserted campus and the nearby woods were strung with yellow tape as investigators scoured them for evidence.
The shooter appears to be of college age and "blended in well" on the campus, Utah Public Safety Commissioner Beau Mason told reporters.
The shooter has not been publicly identified, but the FBI circulated grainy images apparently taken from security cameras showing a "person of interest" wearing a black top, black sunglasses and a dark baseball cap. The top appears to have an image of a bald eagle flying across a US flag.
Ammunition found so far appeared to have been engraved with messages, the Wall Street Journal reported, but people familiar with the investigation told Reuters the engravings and their meaning were still being analysed.
Kirk was co-founder and president of the conservative student group Turning Point USA and his appearance on Wednesday was part of a planned 15-event "American Comeback Tour" of US college campuses. His killing stirred outrage and denunciations of political violence from Democrats, Republicans and foreign governments.
Trump said he would award Kirk the Presidential Medal of Freedom, the nation's highest civilian honor.
Vice President JD Vance canceled his trip to New York to commemorate the attacks by al Qaeda on September 11, 2001, and instead will travel to Utah to visit Kirk's family, a person familiar with the situation said.
Kirk, who was married and the father of two young children, published his most recent book last year calling for a "Right Wing Revolution" and had just returned from an overseas speaking tour in South Korea and Japan.
