  By applying a sudden perturbation to one of the components of a mixture of
two quantum fluids, we explore the effect on the motion of the second component
on a short time scale. By implementing perturbation theory, we prove that for
short times the response of the second component is fixed by the energy
weighted moment of the crossed dynamic structure factor (crossed f-sum rule).
We also show that by properly monitoring the time duration of the perturbation
it is possible to identify peculiar fast spin drag regimes, which are sensitive
to the interaction effects in the Hamiltonian. Special focus is given to the
case of coherently coupled Bose-Einstein condensates, interacting Bose mixtures
exhibiting the Andreev-Bashkin effect, normal Fermi liquids and the polaron
problem. The relevant excitations of the system contributing to the spin drag
effect are identified and the contribution of the low frequency gapless
excitations to the f-sum rule in the density and spin channels is explicitly
calculated employing the proper macroscopic dynamic theories. Both spatially
periodic and Galilean boost perturbations are considered.
