I'm not a criminal master, but I'd like to add that some areas also prohibit certain types of ammo, so they use, "Ball" Ammunition, it passes through the target cleanly. I am very sorry if I seem like that douche-bag who corrects a teacher, and ends up being wrong.
Glad you brought that up too, State regulations and department policy can also affect what ammo type is used. The concept of SWAT came around in the 1960s in Los Angeles with LAPD. The police departments of the time had limited access to equipment back then. Semi-automatic handguns were becoming more common and pump actions shotguns had been common for a long time. Some departments had bolt actions rifles, others had access to M1 carbines, but ammo stocks for it were running dry. There were AR15's being marketed toward the police, but departments avoided them due to political reasons.
The SWAT of the 80s are 90s had MP5s, some AR-15s, Demiled CAR-15s. As far as ammo goes, there was only one type of 9mm you could get, .223/5.56 FMJ, various ammo types for bolt action rifles, and 12 gauge buckshot or 12 gauge slug. 9mm is one of those ammo types that is said to go through people, .223/5.56 can go through people, usually tumbles on impact though. There as also .45 and .40, but not every department uses those and .40(10mm Auto) was unpopular at the time due to the wear and tear it causes to handguns.
Since the late 1990s AR-15s have become more common and have taken the place of shotguns and MP5s. A lot of it also had to do with the war on terror and grants for counter terrorism, basically everyone wants their cool black tacticool rifle now. New and improved ammo has become common as well along with soft point ammunition.