Image Sgt. Graham Agnew during his shift. “There’s a differentiation between speaking to people and speaking at people,” he said. Credit... Kieran Dodds for The New York Times

At one point, a woman fell headfirst onto the street, lay there a moment and got back up and smiled. As Sergeant Agnew explained, the officers were employing a form of “broken windows” strategy — an idea, like others, “we stole from America,” he said. Their aim was to spot the drunken, early on, to help them home or ticket them for breach of peace, before things led to fistfights, car accidents or other violence later.

(“Let’s nip it in the bud,” Sergeant John Harris had said at roll call, “if there is someone carrying on.”)

As recently as 2007, Glasgow was one of the most violent cities in Europe, said Sir Stephen House, who formerly led a police district that covered the city and who stepped down last month as the country’s top officer. There were about 90 homicides that year, he said, a number that has fallen to about 30 today.

“It used to be Glasgow’s No. 1 problem,” Mr. House said. “But it’s not now.”

Riding in the disorder van, Constable Rachelle Logue used another term often heard in New York — “hot spots” — to point out where crime persists: A street where hand-to-hand drug transactions take place; a poorly lit “red light district” where prostitutes roam, though not as robustly as in the past.

Around 40 gangs run on the streets, she said, often in the public housing complexes, known in local slang as “schemes” that dot the city. But the gangs consist mostly of teenagers and are not known for settling fights by shooting. Few guns, said Constable Brian Sexton, who was driving the van, made a big difference for policing here.