From my limited experience...
A320 - no CG management, other than trying to get loaders to get CG towards aft end of envelope. In that BA are also trying to sell "premium" seats, that are towards the front, slightly contrary motives.
A340 (from a previous life). At 25K' on the climb, fuel is pumped rearwards into tailplane tank. The moment you hint towards aircraft you are going to land soon (Dest, or Divert), it empties said tank (I think at ~2hrs to landing?)
Also, noisy red flashing awkening warning "Aft CG Limit" that we don't get on A32x. Normally went off towards end of cruise, and maybe when there was a "congregation" in rear galley. At this point, aircraft started furiously pumping the fuel back forward again.
So - yes, Airbus do try to get aft CG since it is more efficient. Sounds like safety and regulation allow a greater extent of this in cruise, than for TO/Ldg. However, I dount it gets anywhere near the "unstable" regime of modern fighters - just "less stability" than permitted without these protections.
NoD