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 ...

Yeah, grew up in France and I was the only one of my friend who had not modified my scooter. It would litteraly stop accelerating past 50 km/h.

I am not an expert, but in Denmark I believe they literally "drilled" the combustion chamber bigger and replaced the pistons (among other modifications).

Back in the early 90s when this was something my friends were into, several of them had Yamaha 4 gears capable to going over 100kph (60mph) while the legal limit was 30kph (~19mph).

Actual 2 stroke mechanic here with a old tomos. How? It's a literal lawnmower engine. You order up different pistons and head and bada bing put it together you have 75ccs. Kits are around 200usd. Check out treatland a large mopey seller on the west coast

I still recall my first mod on a small motocross 50cc 2 stroke motorcycle I had when I was like 15. The thing had 6 gears, but due to my weight paired by the heavily underpowered engine, I barely could use it up the 5th gear, therefore something had to be done. That was in the early 80s, and the engine was probably a mid 70s design with cast iron cylinder. Some then friends of mine who were into modding for fun told me they could do the work if I bought the mod, so I did and a few days later we were in their garage. The mod consisted in the complete head, cylinder and piston (still 50cc but racing grade chromed aluminium cylinder etc.), plus a new carburetor (from 14/9 to 22 iirc). After the mod was done it took a while to turn it on for the first time, but then after some tuning it became reliable and I could get back home. I spent a couple days driving it, then the engine seized. Having done anything wrong, I called those friends to ask explanations, but as predicted they weren't collaborative, so I bought a second mod (reforming the chromed aluminium cylinder would cost more back then) and this time mounted it by myself. Turns out the old one broke because one of the G-clips on the piston was not properly secured and fell out of its receptacle destroying the cylinder chamber. The new mod, although performed by a complete novice, lasted some good years until the day some bastard stole my motorcycle.

In the end, it cost me all the money I had saved, but was a fun and instructional experience nonetheless, also for teaching me that some friends when things go wrong often stop being such.

So called "big bore" kits are common for 50cc scooters. You replace part of the engine block and sometimes the cylinder head. 50cc is pretty small when you thing about it so you don't need a massive change to get a performance boost.

Here's one online with a photo https://www.amazon.com/Glixal-ATMT1-011-Scooter-Cylinder-Reb...

Most of the time all you need to replace is the cylinder and piston. For two-strokes, this is often a very simple task. It might be as easy as removing four bolts and a circlip.