Region play is asynchronous. Cars may enter or exit the city, but only in a few select cases does a car actually move between cities. Here are the details on police interactions.
If criminals are winning in your city, police in other cities will respond even if the other cities do not assign those vehicles to you. The response will take the form of cars generated for your city, which can arrest criminals and remove them from your city. These criminals do not go into the local jail.
Since the cars and criminals are generated for your city, they do not appear in the other city. It all vanishes at the edge of your city map.
I started a private region and claimed all the city spots. Then I started the first city and brought it up to 100k population, with a police precinct and an all blue police coverage map. I closed the game, re-opened and started the second city. When criminals appeared in the second city, police responded from the first city. There was no remote assignment of these vehicles - I hadn't opened the first city to perform such an assignment. I had no jail at the time - the police left with the criminals.
Later, the second city grew to 150k population, with a precinct and a blue police coverage map. At some point, the first city swapped from sending helpful police to sending arsonists and murders - even though it hadn't been opened since starting the second city. I remote assigned 4 cars from the second city to the first city. Later, I got a message that the crime in the first city was reduced, and the arsonists/murder in-flow stopped. I doubt that the first city was actually affected by my remote assignment. I suspect that, instead, the relationship between first and second city became crime free.
All of this happened with the first city closed.