That thing is off screen development. See, in the Saturday morning cartoons you need to keep ratings, and they try to do that by keeping the show action packed. As such the episodes focus on the times when that action is there. Batman dealing with the Joker's new crime spree, Spider-Man trying to find out what Hobgoblin is up to this time. That sort of thing. However, there is still progress going on in the background, things we don't see. Usually related to sub plots, things going on with the secret identity and stuff like that, or minor things that aren't necessarily super relevant to the going on of the show, but can be important for the characters inside it.
Batman Beyond did this particularly well. Not many episodes at all focused on Terry's personal life, but they touched on it, and using little lines and things they communicated volumes. We know Terry is regularly up all night (and gone early) being Batman, because in one episode when he is home in the morning his brother comments that he is a stranger, and his mother mentions it being nice to actually see him. We know that he is sacrificing a lot of time with his girl friend, because of little comments about his "job" and how rare it is for them to actually go on a date.
None of this is big news on the shows, but it can set up dramatic moments for later in the story with just the little maintenance touches. Time goes on when we're not looking, things happen, the status quo is maintained, but the status quo also has impact on things. The status quo is that the main character spends the vast majority (if not all) of their free time fighting crime, making them late for appointments (if they even show up) and practically a ghost in their friends' lives because of their commitment to what is going on. It's not harped on, it isn't explicitly detailed, we just know it from a simple line here, a comment there.
You can bring this into your games as well. Have down time in between adventures, don't just have each adventure happen right on the tail of the previous one, give some time in between. Give the characters time to grow and develop, practice some things. Give them time to establish a status quo, so we can see where attention is focused when everything isn't going crazy, that way we can know what is up when things do go crazy. You don't need to even do much to explain it either. Give the characters a week of down time, and ask what they're doing. What they do on a normal day. It doesn't need to be super specific, just the general facts. "I wake up, shower, go to school, do my job at the Daily Bugle, and then spend a few hours web slinging before heading home for dinner and homework. If I come across crime over the course of my day, I'll get away, change, and play hero" this is Peter Parker's "normal" schedule. From that we can extrapolate, especially since he lives in New York. Now we can use the things mentioned above to show how relationships are becoming strained, especially if he has a girl friend at the time.
In Fantasy games with more "historical" technological means this may be a bit harder, but even still you can find it. One of my character's daily schedule is "Wake up an hour earlier than everyone else and get ready for the day, morning routine (prayers, run, breakfast), daily duties, evening routine (prayers, run, dinner), training, bed about an hour before anyone else.". Now, some of that is set down by the culture and all. In fact up until "training" and excluding the "hour before everyone" lines about sleep, any character in that game would have the same daily schedule. However, aside from the set stuff, the only thing my character does in general is train. Other characters do other things. One person spends time with his wife, reads tactical treatises, and tries to get to know his troops better. Another does a bit extra duty for his superior and personal improvement, trains a bit, then joins some of the troops for a drink and baudy tales. Everyone is different in this, even with the remarkable similarities. The GM can, and does, use this at times to also poke at the characters as well. Can you guess which of those three characters doesn't have much of a relation with the troops around them aside from "something to be feared, respected, and/or admired depending on the situation"? I thought so.
So, find out what your characters do in their down time. Re-enforce it, use it, break it when you need to to get them to do things. Have fun with it. Among other things, you may be amazed at how much more developed characters become when 3 years of OOC gaming is actually 6 years of IC, as opposed to 6 months of IC. So like I said, have fun with it.