I've been loosely keeping track of this for years, in the back of my mind, and decided to set these down in type:
1. Largely repeated plots with new settings and villains. In one series, two different baddies in two different books targeted the hero's wife out of lingering obsession (one from early childhood, one from college).
2. Growing lack of interest in key, recurring secondary characters from the first few books. Often these people just disappear. One particular kiss of death for a series, in my opinion, is when a recurring antagonist (a dirty cop in the same precinct, say, or a jealous and hateful superior) is killed or otherwise disappears.
3. Sexual tension between main characters leads to consummation, leaving little going forward between the two but contrived domestic dramas (or lingering, accusing, guilty glances that sometimes drag out for books and books).
4. More room is given to developing the character of villains.
5. Abrupt shifts away from the main setting (protagonist is transferred or changes careers).
6. The main characters stop making forward advances in their lives, and instead concentrate each book on fighting a new or recurring evil. The books become almost 100% plot-driven.
This isn't true of all series mysteries, of course. Many die a natural death, and others chug along cheerfully in high gear to this day. But I'd read enough series that became self-perpetuating for their own sake, over about 30 years of serious adult mystery reading, that these commonalities have made themselves apparent.
Right now, I'm noticing one of my favorite series develop several of these traits (most notably 4, 5 and 6) in its ninth and tenth volumes. I'm really hoping the eleventh, coming out next spring, pulls the series out of its gentle tailspin ... but history, in my experience, doesn't support that outcome.
Any additions or quibbles?