IF League were to make a movie or Netflix series, I think it'd have to mainly revolve around...and I can't believe the words are coming out of my fingertips...Ezreal.
He's one of very few champions who literally roams the entire map, pretty much endlessly. Other candidates that could fit that role are Ryze (trying to prevent another Rune War), Lux (only after her exile), or maybe Amumu (but that would probably get very depressing)
Alternatively, they could do it sort of like Robot Chicken, where they just jump all over the place...but I really don't think it'd do them justice like that.
Or they could make up a character or two that roam around, finding themselves in all sorts of strange situations, trying to do something of their own, in which case, me and Voidling each have our own perfect fit in a couple of all regions concepts trying to unite Runeterra against the Void threat. I've already got a collab going that's sort of going all Doctor Who on timelines to bring these guys together...but for the sake of a Netflix series, I'd go with Voidling's concept, as it's a lot easier to actually work with than mine. :P