Just saying

They do. LoL just has archaic spaghetti code where if you get CCed between the initial cast and the end of the cast delay it goes off anyway. Its really shitty old coding that they were too lazy to fix so now it's a "feature"

th...they do(?)

They do. They just don't stop abilities that are buffered before the CC lands like Ezreal's E.

They do prevent movement abilities- like other people have stated, if the movement ability is cast before the snare, the snare effect moves with the enemy.