In all cases the relationship between $u_o(x_o)$ and $u_f(x_f)$ is given by a Fourier transform. My question is, when is the same true for the relationship between $u_f(x_f)$ and $u_i(x_i)$?
Strictly speaking, it's not true that in all cases the relationship is an exact Fourier Transform. Since we're dealing with electric-fields, unless the object is exactly one focal length away from the lens all other cases will have a quadratic factor that needs to be dealt with, unless you're interested in the purely incoherent case.
Also, I'm not sure if I understand your question regarding relating uf and uo.