OK hear me out. I have a 1990 Roller LA 360 out of a wrecked Ramcharger that's in my Duster for now, I did some research into the roller LA's and Magnums awhile back.
From what I can tell Jeeps never had the roller LA, they all had Magnums from 1993 on.
The regular (XJ) Cherokee's and Wranglers only had the venerable 4.0L straight six (saw many of those engine go past 300,000 miles with no issues) as the biggest motor option. (we don't EVER mention the GM 2.8L V6 they put in the early XJ's)
The (ZJ) Jeep Grand Cherokees didn't come out until 1993, by then LA Rollers were phased out and they started using Magnums (but only the 318 for Jeeps), the 360 magnum option for the Grand Cherokee's didn't come out until 1998. The 4.0L were still the standard motor for this platform.
You had the (SJ) Wagoneers that were out until 1991, as mentioned above in it had the AMC 360. (AMC 401 and the 4.0L's precursor the 258 straight 6 were the other AMC engine options offered throughout the years)
And yes Roller LA's did appear as early as around 1985 but only on 318's. As for the roller LA-360's the absolute earliest I've seen them was on a late 1988 Dodge Ram Van (build date 05/88), and yes it was the original engine.
Viable sources for the Roller LA-360 are usually Rams, Van's and Ramchargers from 1989 (maybe 'some' 88's)* model year to 1993. Also the occasional "let's put a 360 in a Dakota or M-Body" project.
Also note (but not related to your question): M-bodies, 2WD Dakotas & some vans are also great sources for the A999/A998 transmissions.
*Emphasis on the some, when I used to go to the salvage yards regularly from what I've seen on late 1988 model years they will have both, but the roller LA360 seems to be more rare.