This thread seems to have gotten a wee sluggish, so let me toss this out. It might fit better on a new thread . . . but we shall see.
I'm close to finishing the first story for my first "series" character, and it all feels fine--but my usual approach is to really root the story in some kind of character change, be it positive or negative. Don, the main character in "A Demon Once Removed," for instance, gets this opportunity to figure out why he's such a blamer and to do something about it. He seems to take, at least, a first step toward a better relationship with his son.
Series characters certainly can change, and Buffy is a good example. She got darker as a she got older. But her change is so gradual, and I doubt that my series will last for seven years.
Typically, though, series characters remain the same. Think about Sherlock Holmes, for instance. He's always . . . you know . . . Sherlock Holmes.
So if character change isn't at the heart of the story with a series . . . what is?