President Trump has sent National Guard troops to several American cities, mostly heavily Democratic ones, this year as a way to help protect federal buildings and personnel amid protests against his immigration enforcement tactics and to address what he claims are rampant levels of crime.

Since June, Mr. Trump has sent soldiers — often over the objections of state and local leaders — to cities including Los Angeles, Washington, Chicago, Memphis and Portland, Ore.

The mobilizations represent a departure from the traditional role of the Guard, a state-based military force whose domestic deployments have historically involved humanitarian assistance. And they have prompted lawsuits across the country, several of them accusing the Trump administration of exceeding its legal authority.

Here’s how courts have landed on those cases so far.

Los Angeles

In June, the Trump administration sent nearly 5,000 troops, from both the National Guard and the U.S. Marine Corps, to the Los Angeles area, citing a need to quell protests over immigration raids. Gov. Gavin Newsom of California filed suit on June 9, accusing Mr. Trump of an “unprecedented usurpation of state authority and resources.”

On Sept. 2, Judge Charles R. Breyer of the Federal District Court in San Francisco ruled that the National Guard troops were being used illegally. He wrote that the Trump administration had violated the Posse Comitatus Act, an 1878 law that makes it illegal to use federal troops for domestic policing in most circumstances.

The ruling barred the federal government from using troops anywhere in California to engage in “arrests, apprehensions, searches, seizures, security patrols, traffic control, crowd control, riot control, evidence collection, interrogation, or acting as informants.”