The case deals with Temporary Protected Status for hundreds of thousands of Haitians and Syrians but could have implications for more than a million from troubled nations.
Prosecutors accused Rubén Rocha Moya, the governor of Sinaloa state, and other Mexican officials of a yearslong conspiracy to protect the powerful cartel.
Republicans put down a right-wing revolt to push the measure through the House, but it faces changes in the Senate that could delay its final passage past a Friday expiration.