On my guns there is a small but noticable change in zero, when scope swapping.
Case in point. Just bought a Howaby. That means another round of musical scopes. Moved the Mueller from my M700 Tupperware Special, rings bases and all. Didn't need to adjust the rings on the Mueller to fit the Howaby.
Bought rings and bases and robbed the 4x Buckmaster form the BC and put it on the Tupperware Special.
Even though the Mueller was bore sighted and dialed in to the Tupperware Special and I didn't need to loosen the rings to fit the scope to the Howaby, once on the Howaby, the Mueller was low and to the right. This is using the same rings and bases off the Tupperware Special.
So, if you play musical scopes, you will need to do a cursory bore sight and pop-off a few rounds to dial things in.
Do a search. I got very good instructions on how to mount, and boresight a scope. No fancy gizmo's needed.
ZM
PS. For the handi-holics. After robbing the scope off the BC, I put a Williams FP on it. The BC just felt wrong with the scope sitting on it. Things are much better now.