  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. We show that the response is fixed by the energy
weighted moment of the crossed dynamic structure factor (crossed f-sum rule).
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.
