  The mutual drag in strongly interacting two-component superfluids in optical
lattices is discussed. Two competing drag mechanisms are the vacancy-assisted
motion and proximity to the quasi-molecular state, in which an integer number
$q$ of atoms (or holes) of one component might be bound to one atom (or hole)
of the other component. Then the lowest energy topological excitation (vortex
or persistent current) becomes a composite object consisting of $q$ circulation
quanta of one component and one circulation of the other. In the SQUID-type
geometry, the value of $q$ can become fractional. These topological complexes
can be detected by absorptive imaging. We present both the mean field and Monte
Carlo results. The drag effects in optical lattices are drastically different
from the Galilean invariant Andreev-Bashkin effect in liquid helium.
