I believe all the ships now offer 'preschool fun' within the first couple of days.
It a time when the club is only open for 3-5 year olds to be dropped off for 30 minutes to an hour, so they can get used to being left in the space with kids their own age and size. They'll often do programmes like 'explore the club', heffalump expedition, puzzle playtime, little einteins during that time for them too.
As for characters in the clubs, it's constant, especially on sea days: detective school with Goofy, aloha and adventures with stitch, pirates with Captain Hook, Sophia, Mickey, Minnie, Donald, Pluto, Cinderella, belle, captain America, Spider-Man, woody, tinkerbell...they'll often come in for a wander too.
Some programmes with characters are often offered in open house which are good for families with young kids such as: mickeys puzzle playtime, always has Mickey and two others (goofy, Pluto, Minnie), and story time with Belle.