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).