Abstract: We use the formalism of Gould and Raffelt [1] to compute the dimensionless thermal conduction coefficients for scattering of dark matter particles with standard model nucleons via cross-sections that depend on the relative velocity or momentum exchanged between particles. Motivated by models invoked to reconcile various recent results in direct detection, we explicitly compute the conduction coefficients alpha and kappa for cross-sections that go as v(rel)(2), v(rel)(4), v(rel)(-2), q(2), q(4) and q(-2), where v(rel) is the relative DM-nucleus velocity and q is the momentum transferred in the collision. We find that a v(rel)(-2) depend ence can significantly enhance energy transport from the inner solar core to the outer core. The same can true for any q-dependent coupling, if the dark matter mass lies within some specific range for each coupling. This effect can complement direct searches for dark matter; combining these results with state-of-the-art solar simulations should greatly increase sensitivity to certain DM models. It also seems possible that the so-called Solar Abundance Problem could be resolved by enhanced energy transport in the solar core due to such velocity-or momentum-dependent scatterings.