Then, as one protester blared a soundtrack of “The Imperial March,” the officers started advancing. Through the acrid haze, they continued to fire flash grenades and welt-inducing marble-size balls filled with caustic chemicals. They moved down Main Street and continued up the hill, where one of the agents announced over a loudspeaker: “This is an unlawful assembly.”

By the time the security forces halted their advance, the federal courthouse they had been sent to protect was out of sight — two blocks behind them…

“If the federal troops are starting to wander the streets, they appear to be crossing the line into general policing, which is outside their powers,” Professor Tsai said.

Homeland Security officials say they are operating under a federal statute that permits federal agents to venture outside the boundaries of the courthouse to “conduct investigations” into crimes against federal property or officers.

But patrolling the streets and detaining or tear-gassing protesters go beyond that legal authority, said David Lapan, the former spokesman for the agency when it was led by John Kelly, Mr. Trump’s first secretary of homeland security.