But when the special counsel’s office closed up shop in May, the Justice Department signaled that its scrutiny of foreign lobbyists would continue. The department put one of Mr. Mueller’s prosecutors, Brandon L. Van Grack, in charge of a revamped FARA unit, declaring that the foreign lobbying disclosure act was “increasingly an enforcement priority.”

How far the prosecutors can push the once-obscure, 81-year-old statute is an open question.

Critics claim that the statute is as vague as it is broad, a problem already evident in the case against Mr. Craig, a former White House counsel under President Barack Obama. Days before his trial was to begin, Judge Amy Berman Jackson of the United States District Court for the District of Columbia dismissed one of two counts of false statements against Mr. Craig, saying the language of the FARA statute was too ambiguous to justify it.

The department’s enforcement drive is also hampered by a vast exemption for lobbyists who are paid by foreign commercial entities, even though business leaders are often stand-ins for foreign government officials who will pay millions to lobbyists to promote their interests in the West.

On the other hand, prosecutors often have an advantage in the courtroom when FARA cases make it that far. Juries are typically unsympathetic to consultants, lobbyists and lawyers who enrich themselves by working for a foreign power but hide their work from their own government. “It is hard to feel sorry for that cohort of people,” said Mr. Sanderson, the defense lawyer.

Consider the most recent case: the prosecution of Bijan Kian, who was accused of failing to disclose lobbying work for the Turkish government. Mr. Kian is a former business partner of Mr. Flynn, the former national security adviser who is awaiting sentencing on charges of filing false documents to Justice Department officials about the Turkish lobbying work, as well as lying to F.B.I. investigators about a different matter.

The federal judge who presided over the trial complained that the evidence against Mr. Kian, who also goes by the name Bijan Rafiekian, was “very, very circumstantial.”

Jurors nevertheless deliberated only five hours before finding him guilty last month of working as an unregistered foreign agent and conspiring to violate FARA, including by submitting false documents. Whether the judge upholds that verdict or throws it out will help define the legal parameters of the department’s new enforcement effort.