It depends on the model of the engine, in some cases the bore can be bored larger (and an increased size piston to match is used) in some the whole cylinder (and piston) is replaced with a similar one but with a larger bore.
But in many cases there was no actual need to increase the size of the engine bore, as more often than not the original engine has much more power and it is limited by external devices (once - think of the '70's/*80's it was common to have the same engine 50 cc with 5-7 HP reduced to 1.5 HP by using an undersized carburetor, a smaller exhaust and even - in some cases a simple washer with a smaller hole in the intake).
I remember how (again this is the '70's) most "legal" mopeds in Italy had a carburetor (a Dell'Orto 14/12, meaning 14 mm diameter engine side, 12 mm air filter side) that was often drilled to 14/14 or replaced by a 18/18 or a 19/19. That was enough, without any other change to go from 1.5 HP (the "legal limit" at the time) to 3-4 HP, then changing addtionally the exhaust it went to 5-6 HP, then lightly enlarging the intake opening and "transfers" in the cylinder (remember these were two stroke engines) and adding 2-3 "transfers" was what made them reach 7-9 HP.
The best (in the sense of funniest) performance kit[1] was one for the Ciao (and similar Piaggio light mopeds that originally mounted an even smaller 12/10) that added a second carburetor (not particularly uncospicuous), example:
https://www.youtube.com/channel/UCenA3H9siBLrpRUWCiLlWHw
[1] intended for circuit racing only ...
