“Bad experiences with police as a black man growing up in Savannah are somewhat of a rite of passage,” he said.

Ms. Smiley, 36, said that the Georgia Bureau of Investigation showed her and other family members the body-camera video a few days after Mr. Boyd was shot. That day, she said, a state detective told her that her son had been armed, and had “wanted to die.”

Ms. Smiley said the video shows Mr. Boyd taking a few steps out of his front door, then falling as the officers shoot. But she and others who have seen the video said they could not make out any weapon in Mr. Boyd’s hands.

“I said, ‘This is the crap y’all brought me up here to see?’ ” Ms. Smiley recalled in a recent interview. Her anger turned to tears. “They didn’t have to kill my son,” she said.

The Chatham County district attorney, Meg Heap, who is white, calls that description of the video “inaccurate,” and said that a grand jury will rely on the results of the bureau’s investigation to determine whether charges are warranted against the officers.

Police and city officials, citing the continuing investigation, declined to comment.

Mr. Boyd’s family said he had dreamed of joining the Marines — an idea his mother disapproved of — or of becoming a police detective. Records show that he was arrested in November 2016 for fleeing the police after riding in a stolen truck with a friend. At the time, Mr. Boyd told the police he did not know the truck was stolen. So an officer asked him why he ran.