Tensions reached a boiling point Thursday as Republicans powered a solution to the border crisis through a Senate committee, brushing aside objections of Democrats who said the extended powers of detention and limits to new asylum claims in the legislation were cruel.

Democrats also complained that Republicans broke Senate rules to pass the bill through the Judiciary Committee, heightening the acrimony.

But Republicans said they were tired of waiting on Democrats to offer solutions to the border crisis and said they wouldn’t let boycotts and lack of bipartisan agreement delay them any longer.

“There is a crisis that’s turning into a disaster. Somebody needs to shut off the flow. This shuts off the flow,” said Sen. Lindsey Graham of South Carolina, the Republican chairman of the committee, who led the push.

He said he had been trying to negotiate with Democrats for seven weeks, offering to accept some of their ideas. But he said he realized they would never accept changes to asylum policy or court rulings that experts say are the driving factors behind the migrant surge.

His bill, which cleared on a party-line vote, would undo a 2015 court ruling that limits the time migrant families can be kept in detention to 20 days. That limit is too short to complete cases, which means migrants are released — and once out in the communities, their cases get pushed back, giving them a chance to put down roots and disappear into the shadows.

Mr. Graham’s bill would allow for longer detention so migrant families can be held until their cases are completed. If they win their claims, they can be released. But if not, they can be quickly deported, he said.

His legislation also would change the asylum system, banning claims made at the border. Instead, Central Americans with valid claims would have to make them at refugee processing centers in Latin America.

He acknowledged it was a major change, but said the situation demands a “time out” so the system can work through an already massive case backlog in immigration courts.

His bill is unlikely to move very far.

Democrats signaled they would try to filibuster it on the Senate floor and, should it clear the chamber, they said House Speaker Nancy Pelosi won’t bring it for a vote in the lower chamber.

Democrats tried to prevent it from getting out of committee, too.

Last week they blocked action by boycotting the committee, denying Mr. Graham a quorum to do business.

He headed that off this week by kicking off the meeting with an immediate motion to set a time for holding a vote.

Democrats said that violated a host of rules and further poisoned the chances for cooperation.

“Why even have rules if they can be broken whenever it’s expedient to do so?” said Sen. Dianne Feinstein, the top Democrat on the committee.

She said Mr. Graham’s bill would remove important protections for migrants and could deny some worthy asylum claims that are being filed amid the surge of bogus ones.

“If the Statue of Liberty could weep, she would,” the California Democrat said.

Senate Minority Leader Charles E. Schumer suggested the better answer was to legalize immigrants who are already in the U.S. illegally — a plan known on Capitol Hill as “comprehensive immigration reform,” which couples legalization with promises of future border security and changes to the legal immigration system.

Mr. Graham said he would consider a broader package than his current bill, saying Democratic ideas such as sending more U.S. aid to Central America would be welcome. He even suggested he would consider legislation to protect “Dreamers” here under protections of the Obama-era DACA policy.

Another option he suggested, should Democrats refuse to bargain, would be to attach his legislation to an upcoming spending bill.

That might violate the deal President Trump and congressional leaders just struck to block big policy fights on the spending bills.

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