  The superfluid drag-coefficient of a weakly interacting three-component
Bose-Einstein condensate is computed deep into the superfluid phase, starting
from a Bose-Hubbard model with component-conserving, on-site interactions and
nearest-neighbor hopping. Rayleigh-Schr\"odinger perturbation theory is
employed to provide an analytic expression for the drag density. In addition,
the Hamiltonian is diagonalized numerically to compute the drag within
mean-field theory at both zero and finite temperatures to all orders in
inter-component interactions. Moreover, path integral Monte Carlo simulations
have been performed to support the mean-field results. In the two-component
case the drag increases monotonically with the magnitude of the inter-component
interaction $\gamma_{AB}$ between the two components A and B. This no longer
holds when an additional third component C is included. Instead of increasing
monotonically, the drag can either be strengthened or weakened depending on the
details of the interaction strengths, for weak and moderately strong
interactions. The general picture is that the drag-coefficient between
component A and B is a non-monotonic function of the inter-component
interaction strength $\gamma_{AC}$ between A and a third component C. For weak
$\gamma_{AC}$ compared to the direct interaction $\gamma_{AB}$ between A and B,
the drag-coefficient between A and B can {\it decrease}, contrary to what one
naively would expect. When $\gamma_{AC}$ is strong compared to $\gamma_{AB}$,
the drag between A and B increases with increasing $\gamma_{AC}$, as one would
naively expect. We attribute the subtle reduction of $\rho_{d,AB}$ with
increasing $\gamma_{AC}$, which has no counterpart in the two-component case,
to a renormalization of the inter-component scattering vertex $\gamma_{AB}$ via
intermediate excited states of the third condensate $C$.
