The results of the autopsy, which the lawyers described in a news conference, came as the F.B.I. announced that it was opening a civil rights investigation into the April 21 shooting, and as Gov. Roy Cooper called for a special prosecutor to take over a case that currently rests with the local district attorney.

It also came amid simmering tension in Elizabeth City, a majority-Black city of about 18,000 people. Residents have been peacefully protesting in the streets since the death of Mr. Brown, who was Black, demanding that body camera footage from the shooting be released to the public. On Tuesday, the city and surrounding Pasquotank County, both already under self-imposed states of emergency, established curfews from 8 p.m. to 6 a.m.

A hearing on whether to release the body camera footage was scheduled for Wednesday morning. The judge hearing the petition, filed by the sheriff’s office, was also expected to consider a separate petition requesting the release of the videos filed by a group of news media outlets, including The New York Times.

Members of Mr. Brown’s family, plus one of the family’s lawyers, were shown 20 seconds of footage on Monday. On Tuesday, lawyers for the family, including Ben Crump, who has represented the families of George Floyd and several other people killed by the police, continued to express anger that they were shown only a snippet of what must have transpired, saying that Mr. Brown had been subject to an “execution.”

Mr. Crump also on Tuesday used the shooting to argue that the culture of American policing remains deeply flawed. He tweeted video footage of a Pasquotank County Sheriff’s Office truck driving down a residential street with deputies sitting in the back, dressed in tactical gear. Mr. Crump referred to it as “the militarized police force rushing to kill Andrew Brown,” adding, “This has become a constant sight across America, the evolution of policing that’s now terrorizing communities of color!”

The video was obtained by WAVY, a Virginia-based television station, through a public records request and shows the truck arriving at a home, followed by shouting and orders to “Get your hands up.”

Deputies from the Pasquotank County Sheriff’s Office fired several times at Mr. Brown after arriving at his house to serve drug warrants. Seven deputies were placed on administrative leave after the killing. Chantel Cherry-Lassiter, a lawyer for the family who watched the 20-second video with them on Monday, said the footage showed Mr. Brown sitting inside his car, hands “firmly on the wheel,” when deputies began shooting.

The shooting continued, she said, as Mr. Brown drove away from the deputies. Lawyers for the family have asserted that Mr. Brown, 42, was unarmed at the time.

The Sheriff’s Office filed a petition in state court late Monday asking that the footage be released to Mr. Brown’s adult son, Khalil Ferebee, after mounting pressure from a range of officials, including Mr. Cooper and the Elizabeth City Council, that it be made public. Under state law, only a judge can authorize the release of body camera footage.

Sheriff Tommy Wooten II, who has faced calls to resign from the local N.A.A.C.P., has said that he supports the release of the footage as long as it would not jeopardize the investigation into the shooting by the North Carolina State Bureau of Investigation.

The findings of an official government autopsy have also not been publicly released. But during the news conference on Tuesday, Wayne Kendall, one of the lawyers for the family, said the independent pathologist’s finding that Mr. Brown was shot in the back of the head supported the lawyers’ contention that Mr. Brown had been shot while fleeing the scene.

“This, in fact, was a fatal wound to the back of Mr. Brown’s head as he was leaving the site, trying to evade being shot at by these particular law enforcement officers who we believe did nothing but a straight-up execution,” Mr. Kendall said.

The F.B.I.’s civil rights investigation was opened by the agency’s Charlotte field office, which will work with federal prosecutors and the civil rights division of the Justice Department, according to a statement.

Mr. Cooper, in a tweet, said the appointment of a special prosecutor would “help assure the community and Mr. Brown’s family that a decision on pursuing criminal charges is conducted without bias.”

Mr. Cooper said his position reflected the recommendation, made in a recent task force report on criminal justice reform, to appoint a special prosecutor to handle all police use-of-force cases.