Yup depends on the champion. Some movements are cancelled (like Cait E) some movements will finish (like Graves E) some movement abilities that weren't yet moving will still "go off" (like Tristana W and Urgot E) which is buffering. But if you catch a Tristana in the middle of her W it cancels like Cait E.
I believe these are the 3 movement situations. Now to decipher which champion has which type of movement ability for 100 champions :) GL new players.
To be fair, Caitlyn E is like the only ability in the game that does this.
Though they did add Yasuo into the game and his E does this, too (he feels new to me, I know he's not).
Otherwise, League has a pretty much fundamental rule that once a spell has been started it will complete unless you're dead or the target's gone. Dashes and blinks are extremely consistent on this front. (Caitlyn is intended sort of but should change, Yasuo is also intended and is more reasonable)