Well, the easy one is that they protag. They start taking action about their circumstances. Even episodic series can have protagonists that are in a weird or difficult situation and they do things. Also, I like them having an arc, where they grow as a person, though that's less necessary for shows I see as episodic. (For instance, it frustrated me that Usagi seemed to take maturity steps back after every arc of the Sailor Moon anime, even the ones without memory loss; some of that might have been her reaction to a sudden easing of stress, though. Which works as a character development moment: she's a teenaged girl and really can't deal with crises all the time yet, but can be frustrating since it sometimes came off as 'return to status quo' instead.)