Most stropping I do is geared toward maintaining an edge that's already in decent shape and usually done with no compound at all, using something like a bare leather belt or a hard-backed denim strop without compound. In other words, I strop only enough to realign a slightly rolled edge or to clean up any fine, loose burrs or remnants of burrs left after resharpening on stones. I also sometimes strop on clean paper laid over an oil stone - I do this immediately upon finishing resharpening on the same stone (usually Fine India), to clean up the burrs and align the edge.
If 'bringing back the edge' is the goal, I'll always use a hone or stone for that. And usually at the finest grit possible, according to my own finishing grit preference, usually in the 320 - 600 range, as with a Fine India stone or a Fine DMT diamond hone. OR, sometimes I use a brown/grey ceramic for that, approximating something in the 1200-grit ballpark.
Any stropping I might do with a compound, I'd only do if I were looking for a more polished finish. Most of the time, I don't want or need that. But for a few kitchen knives, I have used some white rouge aluminum oxide compound (2 -5 micron) on hard-backed denim to polish the thin convex I keep on those. 3-micron diamond compound used on firm/hard wood also works very well for such polishing.