We present the formalism of time-dependent exchange perturbation theory built to all orders of perturbation, for the arbitrary time dependency of perturbation. The theory takes into account the rearrangement of electrons among centers. We show how the formalism can be reduced to the standard form of invariant perturbation theory by “switching off” the re-arrangement of electrons among centers. The elements of the scattering S-matrix and transitions T-matrix and the formula for the electron scattering differential cross section are derived. The application of the theory to scattering and collision problems is discussed. As an example, we apply the theory to proton scattering on a lithium atom, calculating the differential and total cross sections.