Actually there is an easy fix to get them exact, the interpolation of the lattice properties controls the fall off influence. Set interpolation to linear:
What I have done is:
create cube and subdivide
create lattice the same size and subdivisions
set interpolation of the lattice to linea
Make a shape key for the lattice and curve the lattice in edit mode of the shape key.
Put a lattice modifier on the cube.
Now using the shape key influence of the lattice will deform the cube in the same way (without fall off influence)
The goal of a lattice is to deform a mesh so yes if you move, rotate scale or deform it will change the mesh
Edit 3 !!!
Here is a file that I think might be what you are after. What I have done is the same as above then at the end I made a linked copy of the lattice (so now there are 2) you can now move the second lattice around and it will still work.
Lattice.blend (907.1 KB)
Here is the file if you press play you can see the result.
Edit 4 !!**!!!
I made a quick video