WASHINGTON — The Senate on Thursday pushed forward toward passing legislation that would mandate the detention and deportation of immigrants in the country illegally who are charged with minor crimes, as the vast majority of Democrats joined Republicans in clearing away a key hurdle to enactment.

All but eight Democrats and one independent joined Republicans in voting to begin debate on the bill, easily exceeding the 60-vote threshold to avoid a filibuster. Their support signaled that the legislation, which passed the House with bipartisan support earlier this week, is on a smooth path to clearing Congress next week and being signed by President-elect Donald Trump when he takes office.

The move reflected a major shift to the right among Democrats on immigration after their party’s considerable electoral losses in November against Republicans, who campaigned on a promise to crack down on illegal border crossings and carry out mass deportations of immigrants in the country illegally.

The measure is named for Laken Riley, a 22-year-old Georgia nursing student who was killed last year by a migrant who crossed into the United States illegally from Venezuela and who had previously been arrested for shoplifting but had not been detained.

Republicans queued it up as the first of several border bills they hope to revive and enact when they secure their governing trifecta with Trump’s inauguration.

The legislation instructs federal officials to detain immigrants in the country illegally arrested for or charged with burglary, theft, larceny or shoplifting, expanding the list of charges that would subject migrants to detention and potential deportation.

The measure also would give state attorneys general the right to sue the attorney general or homeland security secretary if an immigrant who enters the United States illegally goes on to commit a crime that harms the state or any of its residents.

But many Democrats said they were concerned that the bill could push federal authorities to deport people who might not be guilty of the nonviolent crimes of which they had been accused.

The bill must clear another 60-vote hurdle next week before a final vote requiring a simple majority to pass and clear Congress.