The presence of planets in binary systems poses interesting problems for planet formation theories, both in cases where planets must have formed in very compact discs around the individual stars and where they are located near the edge of the stable circumbinary region, where in situ formation is challenging. Dust dynamics is expected to play an important role in such systems, since dust trapping at the inner edge of circumbinary discs could aid in situ formation, but would simultaneously starve the circumstellar discs of the solid material needed to form planets. Here we investigate the dynamics of dust in binary systems using Smooth Particle Hydrodynamics. We find that all our simulations tend towards dust trapping in the circumbinary disc, but the timescale on which trapping begins depends on binary mass ratio ($q$) and eccentricity as well as the angular momentum of the infalling material. For $q\gtrsim 0.1$, we find that dust can initially accrete onto the circumstellar discs, but as the circumbinary cavity grows in radius, dust eventually becomes trapped in the circumbinary disc. For $q=0.01$, we find that increasing the binary eccentricity increases the time required for dust trapping to begin. However, even this longer timescale is likely to be shorter than the planet formation timescale in the inner disc and is insufficient to explain the observed pre-transitional discs. This indicates that increase in companion eccentricity alone is not enough to allow significant transfer of solids from the outer to the inner disc.