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The Navy, which includes the Marine Corps, has S.P. (shore patrol from ships when in port), MAA (Master At Arms), Security Police (land-based Naval bases), Marine Corps police, and lastly the (Navy) Auxiliary Force that is for back-up. This is how it was pre-2001, I'm sure there's even more now.

If I've never seen it before, it's a new release to me!

"I assume in the real military, the chasers are all MPs, is that right?"
Well, remember this was a story based on conditions about 40 years ago; military protocol is always changing.
Somebody else will have to respond as to how prisoner escort / chasing is handled today.
"I know in the Last Detail, the Chaser were assigned out of some temp pool."
Thanks for reminding me (been 20+ years since I've seen it).
This may be another reason this movie appealed to me; one of so many realistic facets to this flick.
When I got out of boot camp and finished my other job training, etc - I was assigned to guard duty for 30 days (not as punishment) - a transition while waiting for my duty assignment overseas. In addition to guard duty we were also tasked for pretty much whatever odd job that needed to be done.
After my 30 days - I was sent overseas.
When I returned to the states after a year overseas I was assigned to a "transit company" - can't remember what it was actually called, but that's close enough - a "transitional unit" where enlisted folks were housed and fed - waiting for what ever their next unit or dispositional assignment might be (discharge, etc).
I remember getting pulled out of this temp unit for guard duty.
Although I imagine MPs or SPs (shore patrol; Navy military police) did some escort work - there just weren't enough MPs.
Keep in mind that this was the early 70's - a time of drafting people that didn't want to be in the military, let alone go to war; lots of discipline problems; lots of folks going AWOL - lot's of chaser duty; lots of people being escorted back to their units and to the brig. This was also a time of rampant drug abuse in the military and erupting racial tension.
Since there were no where near enough MPs, NCOs were pulled from temp / transitional units to perform those jobs.
NCOs (non-commissioned officers) were and are the backbone of the military.
No matter how dirty and nasty - if you want a job done - look for a sergeant or petty officer. An officer may take credit, but it's the NCO, in the mud and the blood, that makes it happen.

I know in the Last Detail, the Chaser were assigned out of some temp pool. I assume in the real military, the chasers are all MPs, is that right?