One thing I’ve noticed about continuity is how many people don’t really care about it. Another is that people who DO care tend to care a LOT. There was a panel at OzCon, in which I was one of the speakers, about continuity in the Oz books. One topic that came up was that, since L. Frank Baum created such an immersive world, readers want to see it as a place with a consistent history. On the other hand, they’re good enough that we can forgive the occasional mistake. Besides, Baum presented himself as a historian, not all-knowing, even though he wrote in third-person omniscient. If he’s getting these stories from sources in Oz, as he says in his introductory letters, he must be filling in a lot of the gaps himself, because who remembers entire conversations? Then again, in the same letters, he credits certain ideas to his readers, so it’s not entirely consistent even there.
I guess I see him and the Oz authors as a bit more than historians in terms of how accurate their details are, but still fallible. I’m fine with disregarding short contradictory or unlikely statements, but less so with discarding entire episodes.
When a franchise includes multiple media, it’s typical that there isn’t going to be total continuity between them. This is definitely true with Oz, as Baum worked on the scripts for several plays and movies, and they often differed in significant ways from the books. What this sometimes leads to is a sort of lateral borrowing, where a writer will pull elements from one medium into another. When I had seen Ant-Man and the Wasp, I did a little bit of research on the characters, and was interested to learn how many of the characters had the names and certain traits of ones from the comics, but were different in significant ways. TV Tropes has several tropes pertaining to this sort of thing, and I can’t remember which one is which. Sometimes the characters just appear in the other medium with no real explanation, and other times their back stories are changed to fit the different continuity. That happens sometimes with Oz. The stage play of The Wizard of Oz includes a character called Pastoria, the former King of Oz.
Then, when Baum wrote Land, he mentions Pastoria as a former king. He doesn’t appear to reclaim the throne, though, and there’s no indication that he worked as a streetcar conductor in Topeka. When Ruth Plumly Thompson introduces him as a character in Lost King, there still isn’t, but she might have been inspired by the play in giving him a working-class occupation during his exile.
I’ve read part of IDW’s Teenage Mutant Ninja Turtles series (which is to say the ones that I could borrow through Comixology Unlimited), and it does a lot of that sort of thing, drawing in characters and elements from pretty much every previous version of the story, and there were quite a few by that point. I’m mostly familiar with the original cartoon series, and yes, Krang, Bebop, and Rocksteady are all there. But Krang is specifically tied to the Utroms, who were in the original comic series and not in the cartoon (well, aside from Krang himself, and he was never specifically identified as one).
There’s also an attempt to tie together the two origins for Splinter, that he was Hamato Yoshi and that he was Yoshi’s pet rat, by instead making him a mutated rat who’s the reincarnation of Yoshi. I think the trick is keeping in enough continuity for the fans to appreciate, but also not confusing newcomers by requiring too much prior knowledge. It seems like a lot of comics these days rely too much on prior knowledge, even though they’re going to reboot a lot of that stuff eventually anyway.
With Oz, I make some pretty obscure references, but I try not to make understanding them necessary to enjoy the story. How well I succeed, I couldn’t say.