Can I join two different wire sizes on the same circuit?

Avoid it. The breaker or fuse protecting a circuit is chosen to protect the wire size the circuit was designed with. If a thinner wire is added downstream — a common shortcut when extending a point — that thinner section can overheat under a load the breaker considers perfectly normal, because the breaker is still guarding the fatter wire. The thin stretch becomes the unprotected weak link, usually buried in a wall or above a false ceiling. If you're extending a circuit, continue with the same size and grade of wire, make the joint properly inside an accessible junction box, and if the new load is heavier than the circuit was planned for, run a new circuit from the distribution board instead. It costs a little more once, and removes the risk permanently.