  Motivated by recent observations of superfluidity of ultracold fermions in
optical lattices, we investigate the stability of superfluid flow of paired
fermions in the lowest band of a strong optical lattice. For fillings close to
one fermion per site, we show that superflow breaks down via a dynamical
instability leading to a transient density wave. At lower fillings, there is a
distinct dynamical instability, where the superfluid stiffness becomes
negative; this evolves, with increasing pairing interaction, from the fermion
pair breaking instability to the well-known dynamical instability of lattice
bosons. Our most interesting finding is the existence of a transition, over a
range of fillings close to one fermion per site, from the fermion depairing
instability to the density wave instability as the strength of the pairing
interaction is increased.
