He said he welcomed last summer’s protests but condemned the rioting and looting that took place in cities including Chicago, where some stores on the South Side — a drugstore, a gift shop — remain closed.

Over the past year, as the country grappled with the aftermath of Mr. Floyd’s killing, many cities have faced a surge in gun violence that has forced political leaders to reconsider how far they are willing to go to reimagine public safety.

Minneapolis, where many City Council members agreed to dismantle the Police Department before backtracking, has seen a sharp rise in gun violence, with homicides last year reaching the highest level since 1995. The area around George Floyd Square, which has remained barricaded and off-limits to the police, has seen a rise in gang violence, particularly at night.

As mourners gathered on Tuesday, they could not escape the scourge of gun violence. As people somberly gathered, gunshots rang out, sending activists and bystanders running for safety. One girl screamed for her mother as she ran into a corner behind portable toilets, and organizers yelled for children to run to one side.

A short time later, the Minneapolis Police Department issued a statement, saying it had responded to gunshots in the area, and that one person arrived at a nearby hospital with a gunshot wound.

Slowly, activists and others returned to the square, but in a smaller number than before.

As a Black mother, Marietta English, a retired teacher from Baltimore who traveled to Minneapolis for the anniversary, said she was shaken last year when she saw the video and heard Mr. Floyd calling out for “mama.” It could have been her son. From a young age, she said, she taught him how to survive an encounter with the police — keep your hand on the steering wheel, do not reach for anything.