Liberals have been racing to embrace the Occupy Wall Street protests as a left-leaning response to the tea party, but the protesters aren’t exactly returning the favor.

As President Barack Obama praised the movement that approached his doorstep today, some of the protesters outside the White House held signs calling him a war criminal and brandished baby dolls covered in red paint to signify children who have been killed in drone attacks.

“The protesters are giving voice to a more broad-based frustration about how our financial system works,” Obama told reporters during a news conference.

His comments came on the heels of widespread support from labor unions, Members of Congress and liberal advocacy groups for the Wall Street protests against corporate greed that were inspired by the Arab Spring.

“I’m so proud to see the Occupy Wall Street movement standing up to this rampant corporate greed and peacefully participating in our democracy,” Rep. Louise Slaughter (D-N.Y.) said in a statement Wednesday. Protesters on Saturday began a sustained camp in Washington, D.C., called Occupy DC, and similar protests are springing up in cities around the country.

At a solidarity rally Wednesday organized by the American Dream Movement, a coalition of liberal advocacy groups and labor unions that traditionally support Democrats, Rep. Keith Ellison (D-Minn.) told Roll Call that he would be honored to take part in the Occupy efforts.

“They’re upset, they’re frustrated and they don’t have a four-point plan, and they don’t have a detailed plan. So what, you know? What they need is to be listened to,” Ellison said.

Lawmakers have portrayed the protests as validation of their job-creation policies and calls to raise corporate taxes. Not necessarily so, the protesters say. Some of them identify as anarchists and reject the current government — and both parties — as corrupt.

Peter Baldwin, who traveled from Maine to participate in today’s Occupy DC event, said the current system of government should be dissolved. The musical rally originally organized by October 2011, an anti-war group, was overtaken by the Occupy movement, fast spreading to D.C. and other parts of the country. In many cities, liberals have been organizing events with the Occupy name without coordinating with the Wall Street activists.

“This will probably take apart the political system. We’ll start all over again. It has to happen,” Baldwin said, adding that lawmakers are “all bought and paid for.”

The Freedom Plaza protest drew more centrist Democrats, too. Tim Hall, a 20-year-old student in D.C., said he supports Obama and plans to vote for him again.

But activist Matthew Skomarovsky, who has been closely following the uprising in New York, said most of the protesters involved in the movement there are skeptical of both parties.

“There must be an alternative way of building power whose first object isn’t just to elect a Democrat or Republican to Congress,” he said.

The gap between the grass roots and more established groups is not unlike one tea partyers faced in their movement’s early phases. As the conservative anti-tax movement grew, Members of Congress, including Rep. Michele Bachmann (R-Minn.) and Sen. Jim DeMint (R-S.C.), and advocacy leaders, such as former House Majority Leader and FreedomWorks Chairman Dick Armey (R-Texas), rushed to shape it.

Labor unions who joined the Wall Street protesters Wednesday and Members of Congress and advocacy groups who held the solidarity rally in D.C. may try to do the same with the Occupy movement.

Rep. Raúl Grijalva (D-Ariz.), who co-chairs the Congressional Progressive Caucus with Ellison and also spoke at Wednesday’s solidarity rally, drew a distinction between the growing movement and the tea party.

“The tea party movement was built on fear and division, and this movement and this push back is based on a very fundamental idea of fairness and hope,” Grijalva said in an interview.

Skomarovsky said he does not think the Wall Street protesters will succumb to outside pressure.

So far, the organizers have held regular meetings, sometimes with hundreds of people, to reach consensus on even the smallest details. When discussing whether to buy sleeping bags for the winter, some activists raised objections about whether the purchases would support capitalism and undermine their efforts.

“It would be a 180-degree turn to raise money from a bunch of rich people and unions and channel that into the electoral process,” Skomarovsky said. “That kind of process can get a Barack Obama to the White House, but it’s not going to really challenge the power of Wall Street.”

But the movement already appears to be accepting money and help from established groups. Democracy for America, a political action committee started by Howard Dean, sent an email to supporters Wednesday asking them to donate money for sleeping bags to give the protesters.

Jim Dean, chairman of the group and brother to the former presidential candidate, said his goal is to support the movement, not own it.

“We want to help the people that are there and make sure they can stay there as long as they can,” he said. “The longer those folks are there, the more public support they are going to be getting and the better the chances are that we are going to get real reform out of this.”

Jim Dean said he expects the organizers will eventually propose policies to reform Wall Street, but that has not been the case so far. The protesters have issued a declaration that offers a long list of what they don’t like — corporate greed, animal cruelty, colonialism, low wages and discrimination — but little about what they want.

With the tea party, the lawmakers and groups who embraced the movement also gave it definition. Whether the Wall Street protests go down a similar path could determine how successful they are, said David Meyer, a sociology professor at the University of California, Irvine.

“They have to cluster around something,” said Meyer, whose book, “The Politics of Protest,” is focused on political movements. “Even if the first people that called for the movement are not making demands, other people are going to start using their name and making them.”