When the Democratic National Committee recently rejected adding a presidential debate focusing exclusively on issues affectingblack people, it got divergent responses from two groups widely associated with the Black Lives Matter movement.

Campaign Zero , whose agenda centers on ending police violence, quickly embraced the offer for a town hall forum instead and began working to arrange forums for Democratic and Republican candidates. But members of an organization named Black Lives Matter , which first asked for the debates, asserted that only a debate would demonstrate the Democrats’ commitment to their cause.

Black Lives Matter began as a hashtag and grew into a protest slogan — after prominent police killings of blacks over the past year — and became an Internet-driven civil rights movement. The phrase is as much a mantra as a particular organization, with the general public lumping numerous groups under the Black Lives Matter banner, even if they are not officially connected.

Yet amid the groups’ different approaches has been a swirl of political activity. Local affiliates of the Black Lives Matter organization have disrupted numerous Democratic presidential campaign events, pushing the candidates to support policies to end mass incarceration and police brutality.

That organization now has 26 chapters, and one in Canada, that largely set their own direction. In Boston, that has meant protesting the city’s short-lived Olympic bid, which activists said would have been harmful to black neighborhoods. The Grand Rapids, Mich., chapter has held workshops on nonviolent organizing and the prison industry. In St. Paul, Minn., organizers have held rallies to call formore minority vendors at the state fair and protested police shootings.

Yet as the rift over debates versus town halls underscores, the young and sometimes cacophonous movement is struggling to find its voice, as the activists who fly its banner wade into national politics.

Many of those activists and groups favor protest, distrust conventional politics and have no intention of supporting candidates. Others have begun lobbying candidates and elected officials on legislative issues. And still others are hoping to use money to make a difference in elections. Campaign Zero, whose founders gained prominence during protests in Ferguson, Mo., has issued a detailed policy platform on preventing police violence and increasing police accountability. It has also met with Hillary Rodham Clinton and Bernie Sanders to pressure them to embrace its racial justice platforms.

Two groups have started political action committees to back candidates who support ideas espoused by Black Lives Matter activists.One, Black Lives Matter Political Action Committee, started by a St. Louis radio host, plans to raise money for voter education in races for law enforcement-related offices, including for district attorney and judgeships. The second, Black Lives Matter Super PAC , was started by New York activists who hope to raise large donations from celebrities to influence campaigns for a variety of offices.

«At this point, marching and protesting, it’s not going anywhere,» said Tarik Mohamed, treasurer and a founder of the super PAC . «So we’re trying to find new avenues of engaging people for change.»

And, in a sign of its growing influence, the movement is attracting the attention of deep-pocketed Democratic donors, who met with activists in Washington this week to discuss how they can support the budding movement.

The diffuseness and decentralization of the movement is viewed by many activist leaders as a source of energy, with local organizations tailoring solutions to problems in their communities. The broader movement has also fomented a new brand of activism on college campuses, most notably at the University of Missouri. Reacting to what they saw as the administration’s tepid response to racism, activists at the university organized marches, one held a hunger strike and the football team threatened a boycott, forcing the president and chancellor to resign.

«There’s nothing wrong with being decentralized and dispersed,» said Allen Kwabena Frimpong, an organizer with the New York chapter. «The problem is being disconnected. If we are going to build political power, we have to build connections.»

Yet for all the movement’s impact, even some of its sympathizers question whether it needs a clearer organization and more concrete plan of action.

«There has to be a reckoning, I agree with that,» Mrs. Clinton told a group of Black Lives Matter activists after an August campaign event in New Hampshire. «But I also think there has to be some positive thing you can move people toward.»

The name Black Lives Matter was born when Alicia Garza, a California-based activist, used it in a Facebook post after George Zimmerman was acquitted two years ago in the killing of Trayvon Martin, an unarmed black teenager, in Florida. She then teamed with two fellow activists to create the Black Lives Matter hashtag and social media pages. But the movement gained prominence after a white police officer killed Michael Brown, an unarmed black 18-year-old, in Ferguson last year, and the Black Lives Matter founders arranged a national «freedom ride» to Ferguson.

But the ubiquity of the name itself — and the fact that anyone can use it — has caused complications. At some protests, for instance, marchers’ chants have called for violence against police officers. Critics, including several Republican presidential candidates, then equated Black Lives Matter to promoting attacks against the police.

«I don’t believe that that movement should be justified when they’re calling for the murder of police officers,» Gov. Chris Christie of New Jersey said recently on CBS’s «Face the Nation.» He repeated the charge in the fourth Republican debate last week.

But leaders within the movement said that they reject violence and that anti-police chants are the acts of individuals, not the movement.

«It’s like saying, ‘Because the Ku Klux Klan calls themselves Christian, Christianity has a problem and needs to answer for the Ku Klux Klan,’ » said Kenneth Murdock, who hosts a political radio show in St. Louis.

Mr. Murdock, who has worked for politicians in Missouri, started the political action committee over the summer. He declined to disclose how much money he had raised, saying he hoped to endorse candidates in local races across the country.

The Black Lives Matter Super PAC, in addition to contributing to campaigns next year, hopes to capitalize on new technology, such as virtual reality software, to help people understand experiences like solitary confinement, Mr. Mohamed said.

Members of the Democracy Alliance, an influential club of liberal donors, met on Tuesday with groups allied under the Black Lives Matters banner — including ColorOfChange.org, Black Youth Project 100 and the Black Civic Engagement Fund — to discuss possibly directing funds to the movement, said Leah Hunt-Hendrix, an alliance member. The organizations represented only a sample of the groups that donors wanted to shed light on, she said.

«It was just a really real conversation about the complexities of funding movements and the need for more infrastructure, especiallyblack-led infrastructure,» said Ms. Hunt-Hendrix, who has inherited wealth from an oil company her grandfather started.

Specific funding commitments were not made, she said, and it would be up to individual donors to follow up with organizations they want to support.

«We don’t want to raise expectations that this is a secretive group of donors hoping to raise tons of money,» she said.

DeRay Mckesson, a founder of Campaign Zero, has been focusing much of his energy on trying to organize Black Lives Matter-themed forums for Democratic and Republican presidential candidates. Town hall forums are more in line with the spirit of the movement, Mr. Mckesson argued, because they allow ordinary people to ask questions. He said he has been in negotiations with television networks to broadcast the forums.

Mr. Mckesson got support for the town halls from an unexpected source recently when Senator Rand Paul of Kentucky, who is running for the Republican presidential nomination, said on NewsOne that Republicans would participate «if we were smart.»

«They are drawing attention to issues that need to be drawn to,» Mr. Paul said.

Some activists have criticized Campaign Zero as being too focused on legislative remedies for police violence, and were concerned the public would think of them as «a silver bullet» for injustices against blacks, Mr. Frimpong said.

Mr. McKesson has responded by saying, «We acknowledge that this isn’t the radical change required to rethink the system, but we can’t discount or devalue the immediate, practical benefit this can have on people’s lives.»

While Campaign Zero has, among other things, called for investing in better police training and body cameras, other Black Lives Matteractivists are demanding that public funds be used for anti-poverty programs that could drive down crime.

When New York City officials sought to add 1,000 new police officers to the force this year, local Black Lives Matter members and other activists argued those funds would be better spent on youth summer jobs, transit access for low-income people, school social workers and teachers. Such programs would address the root causes of crime, the activists argued, but the City Council voted for adding more officers.

Many members of the Black Lives Matter organization also continue to promote protests of presidential candidates in addition tomore conventional political activities, such as voter registration or education. The tactic has proved effective with Democrats.

Activists disrupted a Clinton campaign event in Cleveland in August, calling for her to stop accepting contributions from groups affiliated with private prisons and investing in causes that help black transgender women. Months later, Mrs. Clinton said she would stop accepting those contributions. In August, activists took over a stage in Seattle when Mr. Sanders tried to speak, and the senator promptly released a racial justice platform.

«I think that it’s not just about changing hearts and minds, it’s not just about changing laws, it’s about actually changing action,» said Elle Hearns, a strategic adviser to the Black Lives Matter organization who is based in Ohio.

Khalil Gibran Muhammad, the director of the Schomburg Center for Research in Black Culture in New York and an observer of the movement, said that many Black Lives Matter activists seemed to be advocating change beyond laws, which they say are unfairly applied to blacks. But history, he said, suggested that activists might have to work within the traditional political system.

«They may have to accept that some people are going to have to sit at the table,» he said, «and work this stuff out.»

One Slogan, Many Methods: Black Lives Matter Enters Politics