Demonstrations continued Monday over the death of an unarmed black man shot and killed by a Madison police officer Friday night, as protests in Wisconsin’s second-largest city stretched into a fourth day and reached the heart of the state’s government.

A large crowd gathered inside the rotunda of the state capitol, about a mile from the apartment where Anthony Robinson, 19, was shot after what authorities described as a struggle with a police officer.

Robinson’s death sparked outrage and drew national attention over the weekend, becoming the latest incident involving a black man dying at the hands of the police. Michael Koval, Madison’s police chief, said he understood how it seemed to fit the narrative that has spurred an ongoing conversation over race and policing, but he pleaded with people for patience while investigations are carried out.

A rally on the campus of the University of Wisconsin took place Monday before students marched to the state capitol, and the demonstrations and actions remained calm and peaceful, with no reported arrests or other issues. Many of the protesters who gathered at the capitol building were high school students who came to attend the rally, the Wisconsin State Journal reported. March organizers had posted on Facebook, asking high school students to walk out of classes Monday morning.

Madison police said they were responding Friday evening to multiple calls about a disturbance involving a person who was suspected of committing battery and who was jumping in and out of traffic. An officer, later identified as Matthew Kenny, 45, forced his way inside the apartment where Robinson was; Kenny was struck in the head before firing his weapon, Koval said.

On Monday, Koval wrote a message to the community, apologizing for what he called a “tragic incident” and saying that the entire community was grieving.

“Reconciliation cannot begin without my stating ‘I am sorry,’ and I don’t think I can say this enough,” Koval wrote on his official blog. “I am sorry. I hope that, with time, Tony’s family and friends can search their hearts to render some measure of forgiveness.”

The shooting will be investigated by Wisconsin’s Division of Criminal Investigation, due to a law the state adopted last year requiring an outside agency to look into every officer-related death. Once that investigation is completed, the district attorney will determine whether charges can be filed against the officer, Koval said.

In many ways, the shooting and the response shows the disparity that exists in how different jurisdictions react to police deaths, both in terms of official inquiries and public outreach.

The Wisconsin law requiring an outside agency to investigate is believed by state officials to be the first law of its type in the country. One of the key things protesters have called for over the last year — and, in particular, during the investigation into the shooting of Michael Brown and after the grand jury’s eventual decision not to indict Darren Wilson — was the appointment of special prosecutors for such incidents. The St. Louis County prosecuting attorney was repeatedly criticized as being too close to law enforcement, with protesters and elected officials asking for him to be replaced by a special prosecutor.

Similar calls followed the lack of charges for the officer involved in the death of Eric Garner in New York last summer. Garner died following what appeared to be a police chokehold, and activists and lawmakers said a special prosecutor should have investigated.

New York authorities have offered different ideas for what should follow a police-involved death in place of the current grand jury system. Gov. Andrew M. Cuomo (D) said he wants to appoint an independent monitor for such investigations, while New York Attorney General Eric T. Schneiderman asked Cuomo to let his office investigate and prosecute these cases. Last month, New York’s top judge got involved, saying that judges — rather than prosecutors —should be in charge of the entire process.

Police in Los Angeles are investigating the death of a homeless man shot and killed by officers there. In that city, there is a specific division within the police department created in 2001 that is tasked with investigating any time an officer uses deadly force.

Meanwhile, authorities in Cleveland asked the county sheriff’s office to investigate the shooting of Tamir Rice, a 12-year-old killed by police last year. Cleveland Mayor Frank G. Jackson said the decision to hand the investigation off from a city team was made to establish  “an extra layer of separation and impartiality.”

The Madison shooting occurred just two days after the Justice Department released a scathing report on the Ferguson Police Department, showing the results of an investigation initiated after Brown’s death. The episode also occurred just a day before President Obama spoke in Selma, Ala., to commemorate the 50th anniversary of “Bloody Sunday.”

Obama tied Ferguson and Selma together in his remarks, agreeing that the federal review seemed to evoke the same abuses that were rampant before and during the civil rights movement.

“I rejected the notion that nothing’s changed,” Obama said Saturday. “What happened in Ferguson may not be unique, but it’s no longer endemic, or sanctioned by law and custom; and before the civil rights movement, it most surely was.”

Since Ferguson became a firestorm last year, some police officials seem to have learned lessons based on how public officials responded there. The Ferguson police were roundly criticized in aftermath of Brown’s death last summer for failing to offer much information to the public, waiting six days before naming Darren Wilson as the officer involved.

After the episodes in Los Angeles and other places, authorities have been quick to speak publicly. In Madison, for instance, the police chief and mayor responded to the shooting and spoke publicly, with Koval offering a message of contrition along with some information about the shooting. Madison Mayor Paul Soglin spoke to protesters Monday and has promised residents that they would get answers about the shooting.

This is not necessarily in response to what happened in Ferguson — after Eric Garner’s death in New York, Mayor Bill de Blasio (D) and William J. Bratton, the police commissioner, quickly spoke out, and that happened three weeks before Michael Brown was shot — but it speaks to a potential lesson gleaned from a situation that drew worldwide attention.

Madison’s police chief, in his message Sunday, pleaded with residents not to view the department only through the lens of this killing.

“I realize that in order for us to achieve greater strides in community-based policing, the cornerstone for making that a reality starts with us earning your trust,” he write. “I want that to happen, my department wants that to happen, desperately.”

Todd C. Frankel contributed to this report.

[This post has been updated to clarify that the division investigating deaths involving Los Angeles police officers is part of the Los Angeles Police Department.]