  We study dynamical and energetic instabilities in the transport properties of
Bloch waves for atomic multi-component Bose-Einstein condensates in optical
lattices in the tight-binding limit. We obtain stability criteria analytically,
as a function of superfluid velocities and interaction parameters, in several
cases for two-component and spinor condensates. In the two-species case we find
that the presence of the other condensate component can stabilize the
superfluid flow of an otherwise unstable condensate and that the free space
dynamical miscibility condition of the two species can be reversed by tuning
the superfluid flow velocities. In spin-1 condensates, we find the steady-state
Bloch wave solutions and characterize their stability criteria. We find
generally more regions of dynamical instability arise for the polar than for
the ferromagnetic solutions. In the presence of magnetic Zeeman shifts, we find
a richer variety of condensate solutions and find that the linear Zeeman shift
can stabilize the superfluid flow in several cases of interest.
