MINNEAPOLIS >> The unrest in America’s cities showed no signs of fading today as embattled police forces from Atlanta to Los Angeles struggled to reclaim the streets and as protesters debated the future of the week-old uprising.

Police and National Guard troops continued a heavy lockdown in Minneapolis, where the death in police custody of George Floyd, a 46-year-old black man, ignited the first protests. Gov. Tim Walz announced civil rights charges against the Minneapolis Police Department and said his administration would investigate whether there had been systemic discrimination against people of color over the past 10 years.

An estimated 10,000 people gathered peacefully outside the State Capitol as a Black Hawk helicopter flew behind the freshly gilded dome. National Guard troops applauded, handed out water and sometimes dropped to their knees in a show of support for the protesters.

U.S. troops positioned military vehicles across Washington and a crowd of protesters at least twice the size of the day before gathered near the White House. It shrank after the city’s 7 p.m. curfew, but more than 1,000 protesters remained, facing police officers across a tall chain-link fence erected overnight.

Thousands more gathered in the heart of Hollywood, Calif., this evening as police officers filed out of trucks and formed a skirmish line.

Mayor Bill de Blasio of New York extended the city’s 8 p.m. curfew — which some groups of protesters again ignored a day after looting in Manhattan — through Sunday. He also promised to take action against the “outsiders,” “gang members” and “common criminals” he said were responsible for looting and violence.

“I know we will overcome this. I want to be abundantly clear,” the mayor said, warning: “We will have a tough few days.”

In more than a week of protests, the violence has continued to escalate, and both the police and protesters have been victims.

Officers in New York City and Buffalo, New York, were plowed over by cars and injured. After protests in the St. Louis area turned violent, four police officers were shot and a 77-year-old retired police captain was shot and killed by looters at a pawnshop.

And in Las Vegas, a police officer was in critical condition today after being shot as the authorities tried to disperse crowds pelting them with bottles and rocks.

But protesters, too, have been caught in the increasingly forceful law enforcement crackdowns. In Atlanta, six police officers were charged with using excessive force after video footage showed them stopping two college students in a car Saturday night, firing Tasers at them and dragging them out of their vehicle.

“The conduct involved in this incident is not indicative of the way that we treat people in the city of Atlanta,” Paul L. Howard Jr., the district attorney, said in a news conference today.

President Donald Trump has pushed on states to crack down even harder on the protests, leaving demonstrators debating how to respond to a growing police and military presence in the streets.

As the demographics of the movement have expanded to include a far more diverse population, the goals have also broadened.

Some demonstrators are demanding that all four officers involved in Floyd’s death be thrown in jail, not just the single officer charged last week. “All four,” they shouted at protests. Many others are calling for the entire system of inequality that they view as the ultimate cause of Floyd’s death to be torn down and rebuilt.

Outside the Barclays Center in Brooklyn, a racially diverse crowd of hundreds has gathered nightly, with people chanting against institutions they feel contribute to racial and socioeconomic divides — including law enforcement and capitalism. Those carrying signs cursing Amazon and Jeff Bezos have stood alongside others demanding an end to police brutality.

“What feels different about right now is people are angry for themselves in a way that we did not see outside of black people four years ago,” said Sharhonda Bossier, who began participating in protests affiliated with the Black Lives Matter movement in 2013.

In some cities, protesters worried that violence could overshadow their message and have engaged in self-policing to prevent emotions from escalating.

On Monday afternoon, Floyd’s brother Terrence visited the spot where his brother was killed, near an intersection that has turned into a peaceful gathering spot for protesters. One of the first things Terrence Floyd did was criticize the looting, window-smashing and arson that has accompanied some of the protests.

“If I’m not over here blowing up stuff, if I’m not over here messing up my community, then what are you all doing?” he asked, his voice growing louder. “What are you all doing? You all doing nothing! Because that’s not going to bring my brother back at all.”

Instead, Terrence Floyd, who came to Minneapolis from Brooklyn, urged people to research political candidates and vote for change.

Some particularly uncomfortable conflicts between white and black protesters have reflected the very divide that protesters are hoping to upend.

In Baltimore on Saturday night, Denicia Baker was chanting, “Hands up, don’t shoot!” outside City Hall with a group of about 200 protesters when two young white men wearing black T-shirts and ski masks started kicking and shaking protective fencing put up by the police.

“Stop!” Baker yelled at the men, recording the episode on her cellphone. “When you do that, they don’t go after you. They come after us.”

During a heated exchange, one of the young men responded: “They’re going to kill you anyway.”

Eventually, other protesters removed the two white men. Baker said she suspected they only wanted to stir up trouble.

In many of the demonstrations, white protesters say they have tried to hang back and listen or play a supporting role by passing out water, hand sanitizer and snacks. Some have gone as far as to use their bodies to shield black protesters as the police are bearing down on them.

But some escalations led by white protesters have not been welcome.

“If you show up in these spaces across the country, you are a guest,” said Takirra Winfield Dixon, a former Obama administration official and activist based in Baltimore. “You don’t necessarily have the right to pick up a brick and throw it through a window.”

In Washington, D.C., protesters squabbled on Monday over how to respond as law enforcement wielding pepper spray advanced upon them. Some shouted through the cloud of chemicals that the group should stand strong; others pleaded to fall back for safety.

Later in the night, tensions rose again after Army helicopters descended to rooftop level, kicking up dirt and debris and snapping tree branches in a maneuver often used in combat zones to scare away insurgents.

Some young people wearing masks responded by throwing rocks at storefronts, while others appeared intent on continuing to march peacefully.

In some cases, the debate among protesters over how to respond to the police breaks down along generational lines. Just before midnight Sunday, rumors swirled that the authorities were about to break up protesters near the intersection where George Floyd died. A black woman and two white men, all in their 20s, hatched a plan to set fire to a barrier of garbage containers and wooden palettes to prevent the police from moving in.

“Hey, do you have flammables in the car?” the woman asked the men.

They ran off, returning with plastic bottles filled with liquid, which they poured over the wood and containers. At that point, a group of people hurried over, telling them to stop. One of them was the woman’s father.

“What you’re doing — they understand that violence,” he said angrily. “There ain’t a vehicle that they have that can’t run through fire.”

The woman looked down. “You’re my daughter. I got you,” he said, and she walked away.