Don't get me wrong, this is another case of me being late to the party on a very cool and popular thing, but I've only just recently gotten a chance to start watching the Legend of Korra. Mysteriously, considering it's popularity, there's a lot of things that they've done right. Things you should consider if you ever want to run a continuation game of one of your own campaigns.
Connections to the Past
Let's just get this one out of the way right up front. Frankly, I don't see a real point to doing a "continuation" game if you're not going to give at least some nods to what has come before. People like playing int he same world they've seen for other games for a lot of reasons, but key among those is often the fact that they've got very powerful feelings for stories they've seen told in that world. They've developed connections with the world, the people in it, and the actions that took place. Because of this, you want there to be at least some nods to the past.
How many you have may depend on how far in the future you run the game from the previous one. I'd recommend doing a long enough jump that the next point is possible while still leaving some nods to the previous game. Cameo appearances, descendants, and other links to old favorite characters are always popular, as is anything where you can show the extrapolation of an idea or project that someone put into motion in the big game.
Not Everything Can Be Tied To the Past
Just as important as having ties to the past game is, you also need to have new things. The world needs to be able to stand on its own. Things that weren't featured on screen have to have happened. Most importantly though, there needs to be new places for big threats to come from. Yes, some of them can, and should, come from the old stuff but you also need room for the new. Otherwise everything kind of stagnates around the one time period from which everything that matters came from. (This is, by the way, one of the areas that Korra really excels)
Reinvent/Refocus on the Large Scale
It's one thing to use the same world again, but if you're going back to the same place you also want to try to show a different side of things. If one game focused on a specific city, maybe you focus on a different city the next time. If one game focused on the politics and high society, maybe bring the focus to the criminal underworlds or the more to the grounded side of things.
Like with the allowing for new things this is to help prevent stagnation. It doesn't mean that you can't bring those familiar elements back into place. It doesn't even mean that they can't feature prominently. Just don't have it be the whole thing.
Don't Forget The Transition
Most of these types of games involve a new generation having to take over for the old. The trap here is that you can't downplay the importance, strength, or other abilities of the old. After all, that is what got everyone to where they are, right? At the same time though, you can't forget that the new has to be equally important. If the transition doesn't happen, or if the old generation just continues to do everything, then it can rob the point of the story. Especially in a game when it is likely that the players are playing the new and all of the old generation are now NPCs.