How can I upgrade my Police Station to a Precinct without releasing all the criminals in jail?

Moving from a Fire Station to a Large Fire Station (i.e. closing the small one, building the large one, and deleting the small one once the large one is fully operational) is relatively painless. Moving from a Clinic to a Hospital kicks all the sick guys onto the street and does I don't know what to the injured guys, but it generally doesn't cause much of a noticable problem.

Moving from a Police Station to a Police Precinct releases all the jailed criminals onto the streets, which makes things hairy for several days. There must a better way of doing this than to just turn the old building off and let the baddies loose. Similar to how one can let a Trade Port/Depot sell all its stuff before demolishing it, can one have a station stop arresting criminals but still keep them and rehabilitate them? I tried reducing the number of cars but it looked like it was going to take an extremely long time since you can't reduce the number of cars to zero (so new guys would keep coming in).

Stopping arrests until the jail is empty allows pretty much the same number of criminals as just emptying the jail straight out. I don't think removing cars is the answer.

A police station only holds 10 criminals. Each jail addon holds another 15. You can shutdown or bulldoze the jail addons separately from the police station.

If you bring up your precinct first, then shutdown jails on the station gradually, your city will only face a surplus of 15 criminals at a time. The existence of the precinct should more than make up that surplus.

It sounds like you're doing in-place upgrades for your fire/health/police buildings. As you mentioned, there are ramifications for destroying these older buildings to place new ones on their graves.

Build a new Police Precinct and then leave the Police Station open for several days. I've noticed that the newly arrested criminals get brought to the Precinct rather than the Station.

I won't say its guaranteed without further testing, but it does work when I've done it in the past. You can hedge your bets by slowly closing down jail cells as the number of criminals falls, and moving your cars from the Station to the Precinct. Eventually you can just cut the power entirely, or demolish it.