Police procedurals/dramas are a huge proportion of TV shows, and have been for decades - from episodic shows like the countless Law and Order varieties to genre hybrids like Brooklyn 99.
How did this start? What factors contributed to this type of show being so common? How have audience demand and network programming decisions interacted?
I'm mostly familiar with Anglosphere televison, but would love to hear about the history of cop shows in non-English television as well.
All right, all right, I'll take this one on.
The great grand-daddy of the Police procedural is, of course, detective fiction. Sir Arthur Conan Doyle creates Sherlock Holmes and an (essentially) new type of story is created: one where the major characters are sought out by the story's minor characters, a crime is detected and resolved, at least until the next one. It's a format that has been invented and reinvented many times, and-- long story short-- jumped from popular fiction to radio to television.
DRAGNET is noteworthy as a police procedural series that made the jump from radio to television. At the time-- the 1940s-- other shows were doing the same, mostly soap operas like Guiding Light and As the World Turns. It's hard to reckon anything as the "first" when it comes to detectives solving mysteries, but DRAGNET was an early phenom. The radio episodes are nice, tight mysteries. They have a predictable format:
Ladies and Gentlemen, the stories you are about to hear are true. The names have been changed to protect the innocent. CATCHY THEME MUSIC. A second-person breakdown of the status quo, and a promise that the story will resolve neatly. A call to action: "Your job: find 'im." More catchy music and an ad. Then Joe Friday (you) tells you the date, day of the week, weather, and we're off.
These sorts of stories are infinitely refillable. There will always be another crime to solve, another patient to cure, another case to argue. The creator of Perry Mason (the original novel series) likened his novels to games of baseball: fans of baseball don't complain that the formula is the same from game-to-game, they are interested in the variations that occur within that formula. I think it's hard to fully understand any sort of genre fiction without wrapping your head around that concept.
Moving past DRAGNET and the early days of television, the genre has mostly been defined by someone upending it. KEYSTONE KOPS (silent films about police incompetence) was itself upended by DRAGNET (radio stories about the cops always getting it right). THE ANDY GRIFFITH SHOW (a funny show based in the station) was upended by HAWAII 5-0 (What if we want to get outside and blow stuff up, like on the A-TEAM?) Okay, but what if we want to show the grittiness of urban crime? HILL STREET BLUES. What if we don't want to end every episode in bed? What if we want to do season-wide arcs? NYPD BLUE. What if we didn't make this stuff up, we really shot it, as it really happens? COPS. What if we did a series that was based on true crime journalism, not just detective fiction? HOMICIDE: LIFE ON THE STREET and THE WIRE. What if we did a series that explicitly ignored that, but had access to great actors and locations for a few days at a time? LAW & ORDER. Or, what if, maybe we look at the criminals and their families, and not the cops? THE SOPRANOS. What if the cops are the bad guys? THE SHIELD. What if it was a romantic comedy? CASTLE. What if he's not a crime detective, he's a doctor detective? HOUSE. What if we've got a big crew sitting around in Hawaii because they just finished shooting LOST and all need jobs? Reboot HAWAII 5-0. Ka-BOOOM.
As to your question about the foreign market: it is dominated by the US. It is easier (and more profitable) for the US to buy the rights to a foreign show, reproduce it, and sell those episodes (usually dubbed) to international markets, than it is to market some sort of French police show in Israel, or vice versa. Hawaii 5-0 makes a billion dollars for CBS/Viacom, just in Vietnam.
Cop shows are cheap to produce, endlessly refillable, and sell everywhere. This makes them an easy greenlight for studio executives who want to keep their jobs or get promoted. There are also tried and true practices for marketing them to advertisers and audiences. Imitators are a dime a dozen, but the genre is also an opportunity to reinvent the wheel, and make a fortune doing it.
Further reading/sourcing: The Last Great Ride, Brandon Tartikoff's memoirs. He was the producer of Hill Street Blues, as well as dozens of other hits for NBC, and notably figured out how to game the Nielsen ratings by programming bangers on the night before the family surveys were due. He was a master of manipulating data and getting paid for it, and died too young. Television owes him a debt of gratitude.