It will take time. With Disney classic essentially dead, only Pixar is pushing fully in that direction (quality drama through animation). Dreamworks made a good attempt at it with How To Train Your Dragon (helmed by Disney classic alumni Chris Sanders, co-director of Lilo and Stitch), but their other attempts at drama have been acceptable (Over the Hedge) or awful (Shrek 3). Dreamworks is still pushing irony (Kung Fu Panda, Shrek 1 and 2).
Meanwhile Sony is still pumping out side-kick comedies with Open Season, Open Season 2 and Open Season 3. They made one good attempt at a critter film (Surf's Up), but they're not following that. They did another comedy (Cloudy with a Chance of Meatballs), and I heard it was excellent, but I haven't gotten around to seeing it yet.
The team I'm most interested in is the tiny little effects house that did Happy Feet. They're known as Animal Logic. Their next movie (Legend of the Guardians) looks interesting despite the awful pop song the trailer guys picked, but then, they're trailer guys so who expects better? If Animal Logic works hard on the drama, relationships, plot and character arcs they can do some magic.
Legend of the Guardians trailer: