Summary: Continuing previous work reported in an earlier paper L. M. Burko, A. I. Harte and E. Poisson [Phys. Rev. D 65, 124006 (2002)], we calculate the self-force acting on a point scalar charge in a wide class of cosmological spacetimes. The self-force produces two types of effect. The first is a time-changing inertial mass, and this is calculated exactly for a particle at rest relative to the cosmological fluid. We show that for certain cosmological models, the mass decreases and then increases back to its original value. For all other models except de Sitter spacetime, the mass is restored only to a fraction of its original value. For de Sitter spacetime the mass steadily decreases. The second effect is a deviation relative to geodesic motion, and we calculate this for a charge that moves slowly relative to the dust in a matter-dominated cosmology. We show that the net effect of the self-force is to push on the particle. We show that this is not an artefact of the scalar theory: the electromagnetic self-force acting on an electrically charged particle also pushes on the particle. The paper concludes with a demonstration that the pushing effect can also occur in the context of slow-motion electrodynamics in flat spacetime.