Cheshire = Schroedinger
Hmmnn.. I suspect yer Cheshire cats and yer Shroedinger Cats are actually instances of the same particle (let's call it a felon,pronounced fee-lon) at different points in time. Bit like the thing with neutrinos bouncing between three different states that foxed us for so long re solar neutrino output.
You see a cat (state 1, domestic moggie felon) which then slowly vanishes, leaving a grin behind (state 2 Cheshire cat felon) and then inexplicably re-appears inside a locked wardrobe angrily demanding food (state 3, Shroedingers cat felon).
Consider - the particle has a long life and decays slowly from one state to the other, so it must be massive, and, compared to yer average proton, a felon is absolutely gigantinormous - about domestic moggie sized in fact. The collision of two felons usually causes them to bounce apart, but occasionally the energy involved cause the expulsion of sub-particles, let's call them kitons, which drain the energy of the parent felon, and gain mass as they slow down. Which suggests they are, in some sense, superluminal, maybe its their charm quotient that's superluminal? But it'd also explain how they can have been lounging on the sofa you just got up from, but mysteriously appear in the kitchen ahead of you without having actually overtaken you on the way. It's all classic Quantum, guv, I swear it is.