Democrats are demanding that Republicans slow consideration of picks for the new administration until they can review background checks, as the G.O.P. faces major pressure to quickly confirm them.
President-elect Donald J. Trump has made a special appeal for Pete Hegseth, left, his choice for defense secretary, who is scheduled to appear before the Armed Services Committee on Tuesday.
Most Democrats joined Republicans in voting to take up the measure, which would require the detention and deportation of undocumented immigrants charged with minor crimes.
The action put the measure on track for likely enactment given strong support for it among Republicans, who now control the Senate, and President-elect Donald J. Trump.
Karim Khan, the chief prosecutor of the International Criminal Court, in Paris last year. Congressional Republicans have been trying to crack down on the court since May, when Mr. Khan announced he was seeking warrants for Israeli leaders.
Forty-eight Democrats joined Republicans in supporting legislation that aims to deport more migrants charged with nonviolent crimes, a first salvo in a broader crackdown.
The vote on Tuesday reflected how Republicans in Congress are using their power to revive and pass a raft of border security measures that died during the last Congress in the Democratic-controlled Senate.
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