Just discovered, I don’t have to parent a mesh to a lattice to make the lattice deform it any more!
Thanks to the modifier stack, I can simply apply a lattice as a mesh modifier - COOL!
This means I can now keep the heirarchy of eyelids parented to eyes and do cool cartoony deformations on all of them at the same time!
I can also put the lattice within the heirarchy of an object so it moves with it instead of having to make the lattice the base model!
Another thought - I can use a lattice to make a “zoom to a point” effect inside a wormhole and then fly any number of spaceships into it just by adding the lattice to the ships’ modifier stack.
Thanks Blender Dudes - this rocks!