Growing With the Audience

You were a kid in The Eighties and grew up watching your favorite Merchandise-Driven cartoon but lost interest as you grew older. Suddenly it's The Nineties, and you're bored, flipping through the channels one day, and what do you see? A Darker and Edgier revamp of the show you used to watch! It's good! You get sucked right into it! Fast forward to the Turn of the Millennium, and you hear news that this show is being adapted into a big budget Live Action Adaptation. You go into the theaters, and what do you notice? All the other moviegoers are in their 20s like yourself and probably grew up watching the show like you did. This isn't a coincidence; whoever created the show made a decision to gradually increase the target audience's age as its fans grew older. This trope is one of the biggest sources of Old Guard Versus New Blood trouble around. It's absolutely great for the old guard, but the new blood often feels it just isn't the same if they came in late.

For instance, when Degrassi the Next Generation aired, a lot of old-school Degrassi fans wished the show had stuck to the old characters (who were now adults), while the new Degrassi fans were annoyed that adult characters had their own storylines in a Teen Drama. Later, when the Next Generation cast got too old to stay in High School, the producers were stuck either following them to college and on (which didn't really fit the format) or switching to a new bunch of kids (who nobody cared about). The producers did both and satisfied nobody.

Tropes used in Growing With the Audience include:

Transformers started as a daily syndicated cartoon based on a line of toys, The Nineties brought the Darker and Edgier Beast Wars, and the current decade sees the Michael Bay live-action movies.
J. K. Rowling has stated that she intentionally wrote Harry Potter to encompass more mature and scarier themes as the young readers got a little older for each book.
Warren Ellis wrote a Darker and Edgier treatment of G.I. Joe called G.I. Joe: Resolute, which premiered as a Web Original series. While hardcore current fans did not really appreciate the changes, It did receive positive reviews from casual fans who had grown up with the series.
One could argue that Toonami and Ben 10 did this. With debatable success.
Contra: Shattered Soldier, the Bionic Commando sequels, Final Fight Streetwise, and pretty much the entirety of Prince of Persia in the last decade.
One word: Diniverse. Especially Justice League Unlimited.
John Kricfalusi tried this with Ren and Stimpy Adult Party Cartoon, and didn't exactly get a positive response.
The entire American comics industry has fallen into this over the past 20 years or so, with about 90% of title out there right now focusing on the teen/twenty something male demographic.
Toy Story has been growing with its initial audience through out the entire trilogy, most notable in Toy Story 3 where main-character Andy is set to go to college. No points for anyone who can guess where most of the original Toy Story fans are or getting ready to go to.
Rugrats fits this trope because when it first aired it was a children's show that focused on the exploits of toddlers. However when the show passed the ten year mark, it was revamped into All Grown Up!, aging the protaganists to the status of pre-teens to appeal to the aging original audience of Rugrats.