I've done no-stay MRs to LHR and NRT, as well as short-stay runs to those cities and ZRH, EZE and GIG.
Never had any problems in any of those cities - but occasionally had an a-hole of an immigration agent back here in the US upon return. LAX is usually the worst, while MIA and JFK seem used to the concept of MRs.
Never any checked bags on a mileage run (nor on business trips, as a rule).
Plenty of quick turn MRs in the US - ALB, PVD, JFK, DCA, JAX, MIA, MCO, LAX, SFO, SAN, and SDF, to name a few. Never any problems checking in or boarding, but FAs can be a little shocked to see your face on the return flight just 15 minutes after you arrived.