Attorneys for families of three men killed in confrontations with police are seeking court-appointed monitors to overhaul how the New York Police Department and two other local law-enforcement agencies handle calls involving emotionally disturbed people.

The attorneys, Randolph McLaughlin and Debra Cohen, want the police to create crisis-intervention teams that make mental-health professionals available when officers are responding to disturbed people.

Mr. McLaughlin and Ms. Cohen sued the city of New Rochelle in federal court Tuesday for wrongful death on behalf of the family of Samuel Cruz, 48 years old, an artist and former teacher diagnosed with schizophrenia and bipolar disorder. He was fatally shot May 26 by police who said he was wielding a knife inside his home.

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The lawyers have previously sued on behalf of the families of two other men shot by police, a White Plains man and Mohamed Bah, a Harlem man shot in 2012 by NYPD officers.

The demands for monitors comes two months after a federal judge appointed one to oversee the NYPD's stop-and-frisk program. In Mr. Bah's lawsuit, the attorneys ask that the role of the NYPD monitor be expanded to oversee the agency's policies on responding to calls involving the emotionally disturbed.

White Plains police declined to comment Tuesday, and New Rochelle police didn't return calls seeking comment. Neither New York City nor New Rochelle has responded in court to the lawsuits, but the city of White Plains has asked in court filings that the suit against it be dismissed, citing grounds including officer immunity.

However, officials from those police departments and the NYPD said immediately after the shootings in media reports that their officers responded properly and that their training for dealing with emotionally disturbed people was adequate.

John McCarthy, the NYPD's chief spokesman, declined to comment on the lawsuit's demand for a monitor. He said all NYPD officers are "trained extensively on how to recognize and respond to emotionally disturbed persons."

Officers in the Emergency Services Unit and the Hostage Negotiation Team, which respond to barricaded people, also "receive intensive training from mental health professionals," he said.

The NYPD isn't dismissive of making health-care professionals available to officers in the field. But Mr. McCarthy said they have "logistical concerns with this proposal." The NYPD responds to more than 100,000 emotionally disturbed person calls each year.

At an event last month to mark one year since Mr. Bah's death, several New York City Council members joined a coalition of health-care providers outside City Hall calling for the NYPD to adopt the crisis-intervention team approach.

Democratic State Sen. Kevin Parker, who represents a Brooklyn district, has said he planned to introduce legislation to create a pilot program in New York City.

The so-called Memphis model has the crisis-intervention team as its centerpiece. It requires officers to initially receive 40 hours of specialized training. At least 30% of a police department must receive for the program to work effectively. The trained officers then are deployed as leads in calls involving emotionally disturbed people. Mental-health professionals are on call to advise the officers.

The teams are used by more than 2,600 departments in the U.S.—including some big ones, such as Chicago—said Randolph Dupont, the University of Memphis professor who helped develop the idea in 1988.

The circumstances in all three fatal New York confrontations mirror each other.

Police were responding to calls for medical assistance. The men refused to leave their apartments and were alone. Decisions were made to knock down doors and forcibly take the men into custody. Before shooting, police used nonlethal alternatives such as Tasers.

"In each of the three cases the police have given no explanation as to why they felt there was some need to break the door down," Ms. Cohen said. "There has been no evidence given that any of these three men represented a danger to themselves or anyone else."

Mr. McLaughlin and Ms. Cohen represent the family of Kenneth Chamberlain, 68, of White Plains, who was shot to death in 2011 by White Plains officers who said he was wielding a knife inside his apartment.

That suit seeks $21 million in damages.

The incident involving Mr. Bah, 28, began after his mother, concerned about his mental state and physical appearance, called an ambulance, his mother, Hawa Bah, said.

But police arrived first at the Harlem address.

"'You knocked on the wrong door,'" Ms. Bah said her son told the officers. "'Go away. I don't need the police.'"

An officer then tried to push his way into the apartment, Ms. Bah said, but her son shut the door, locked it and announced he was going back to bed.

Led outside the building, Ms. Bah watched as many officers arrived. She said one carried a shield and another a large hammer. She said she told officers she could talk her son into coming out but was told her help wasn't needed.

According to the police, Mr. Bah came at officers with a foot-long kitchen knife. They used a Taser to stun him, shot him with a rubber bullet and stunned him again with a Taser. He slashed one officer's protective vest and punctured the first-aid kit attached to another officer's uniform.

The lawsuit claims Mr. Bah was shot at least eight times. Police haven't specified the number of times Mr. Bah was shot.

Ms. Bah said she was ridden with guilt that her call for help led to her son's death: "He was just a sick person who needed medical assistance."

Write to Sean Gardiner at sean.gardiner@wsj.com