I am going to dump out some information but not actually answer your question, because I don't feel qualified to speak for all the moderators.
A lot depends on individual circumstances. Moderators on TMC generally have privileges on specific forums (with the exception of the head moderator, plus the site admins). Any moderator can move a thread out of a forum where he/she has moderator privileges...in particular, it's not necessary to be a moderator on the receiving forum. There are a couple of related edge cases however:
1. If individual posts from a thread in one of "my" forums need to be merged into an existing thread where I don't have privileges...this needs either a "global moderator" to deal with it or the co-operation of a moderator on the receiving forum.
2. If I need to edit a post in the thread I'm moving ("Moderator note: This thread was moved from the Model X forum"), or the thread title, I need to do this while the thread is still in one of "my" forums, otherwise I need to ask for help as in #1 above.
In many cases it's obvious where to put a misplaced thread, such as "for sale" postings, or threads that are obviously of regional interest (e.g. California HOV stickers in the general Model S forum). In some other cases it's not as obvious where to put a thread, so I will occasionally reject a report to move a thread because I can't find a better home for it.
There's also a school of thought that many members will tend to stick more-or-less exclusively to the forums for the model of car they own, so they'll tend to look for information there, and make all their posts there.
Some areas of the forums don't have moderators delegated to them specifically, and fall under the catch-all auspices of the head moderator and admins.
Some forums and moderators are more strict about being on-topic (and thus more quick to deal with a report like you mentioned) than others.
Not sure if that's helpful or not!
Bruce.
