  We study the temporally nonlocal contributions to the gradient expansion of
the pair fluctuation propagator for spin- and mass-imbalanced Fermi mixtures.
These terms are related to damping processes of sound-like
(Anderson-Bogolyubov) collective modes and are relevant for the structure of
the complex pole of the pair fluctuation propagator. We derive conditions under
which damping occurs even at zero temperature for large enough mismatch of the
Fermi surfaces. We compare our analytical results with numerically computed
damping rates of the Anderson-Bogolyubov mode.
