Canonical vs localized orbitals - stationary towards changes?

So regarding Koopman's theorem I found the the explanation that you should not use it to find the ionization potential of localized orbitals because, much unlike canonical orbitals they will not be stationary towards small orbital changes. So any minor perturbation will cause a big difference in energy.


Usually I would simply say that the canonical orbitals are the eigenvalues of the diagonalized fock matrix while localized orbitals are not direct solutions of the fock matrix. 


But isn't localization simply another transformation of the matrix that keeps the energy but sort of contracts the density more around bonds or atoms to resemble the bonds or lone pairs we know from organic chemistry? How does this effect the way they behave towards small changes to the orbitals? 


