“Westchester!”

“Fahhh Rockaway!”

“Tel Aviv!”

That one made him perk up a little, inside a local Republican headquarters overstuffed with National Rifle Association hats and “Make America Great Again” casual wear.

“Tel Aviv,” Mr. Giuliani said, noting his own city’s Jewish population. “That’s part of New York.”

Mr. Giuliani, the former New York City mayor and current presidential lawyer, had come to campaign with Ron DeSantis, the Republican nominee for governor here. He had also come to brag a little, in a place with plenty of New York expats and other admirers.

His city, he observed, saw tumbling crime rates on his watch. His mayoralty had made him the kind of figure Republicans called upon as a surrogate in faraway states.

What, then, of Mr. DeSantis’s Democratic opponent, Andrew Gillum, the mayor of Tallahassee?

“He was mayor of a big city, good test. And he failed completely,” Mr. Giuliani said, citing crime issues in Tallahassee, including a fatal shooting on Friday. “My goodness, the guy would be a wreck if he cared about all the murders in the city. He’d be going around like this.”

Mr. Giuliani made a face that mayors are supposed to make, presumably, when they care.

Mr. Gillum has framed the violence as a function, largely, of N.R.A. influence in the state.

Mr. DeSantis would have none of that. Taking the microphone from Mr. Giuliani, he suggested that Mr. Gillum — who has chafed at his opponent’s tendency to call him “Andrew” in debates — had not earned a fuller title.

“In the military you’re entitled to be called by the rank only if you served honorably in that position,” Mr. DeSantis said. “And the fact of the matter is, he has not served honorably as the mayor.”

— Matt Flegenheimer