I’m not sure what they do with all that mail that’s not addressed to me but I don’t really care. The point is I don’t have to see it.
This, along with the other tips in these comments (such as calling customer support lines and requesting removals) means I’m down to literally about one piece of junk mail per month these days.
My wife had the actually great idea to leave a pen in the mailbox for this purpose. It works really well.
In any case after a few years I just toss their mail in the trash, junk or otherwise. If they don't care to file change of address forms then I don't care either. I don't think this is technically legal, though.
If it's not addressed to you at all then you should probably return it of course.
Recently I got one for my brother, who moved out a couple of years before I did.
For rural routes the carrier is supposed to put a card requesting the names of addresses when someone news moves in. I’m not sure what they do with this info but I think they can potentially do something with mail not addressed to your while casing their mail.
I overheard someone being told to put a card in the box about the issue and if that didn’t work to come back and they would enter a COA. But my own post office didn’t seem to know that option and just said to tape a paper on the box door saying “only deliver for X.”