With the dogs; Most of the time they are adopted out. Dogs in Japan are incredibly expensive, so offering a purebred dog at a low price, even if they are a bit past the tiny puppy stage, can usually find an owner. For those that don`t... They either move to different cafes or go on to places like this;
IPCわんわん動物園 ハッピーインフォメーション
Where tons of dogs accustomed to being messed with by countless people wander about getting attention.
For the cats - kitten cafes are not nearly as common as full grown cat cafes (at least in my observance) so I would assume they just move to a cat cafe.