Point of impact changes with load or bullet changes

As a beginning reloader I have a question for those with more load development experience then me. When changing loads does the center of the group move in the vertical plane only or does it move both vertically and horizontally? How about bullets? Does a change in bullet weight move the center of the group vertically only?

As a side note I just realized that electronic targets and a Ransom Rest would be a dream testing set up. I've always liked the center of group indication when shooting electronically score targets.

So far my Ransom Rest experience was to see if my 22 held the X ring at 50 yards and my 45 the 10 ring with my competition ammo. Since they did I stopped testing. I'm just posting the question as I've managed to collect several boxes of bullets and 22 ammo that is not my regular competition stuff.

I load 4.2 n310 w/ 185jhp for 50yards and 3.8 n310 w/ 185jhp for 25yards i have to come up 8clicks(aimpoint..) and left 5

Both sometimes , lead using same bullet most will be just elevation, FMJ can be both or different brand lead can be both.