A model describing the ciliary driven flow and motion of suspended particles in downstream suspension feeders is developed. The quasi-steady Stokes equations for creeping flow are solved numerically in an unbounded fluid domain around cylindrical bodies using a boundary integral formulation. The time-dependent flow is approximated with a continuous sequence of steady state creeping flow fields, where metachronously beating ciliary bands are modelled by linear combinations of singularity solutions to the Stokes equations. Generally, the computed flow fields can be divided into an unsteady region close to the driving ciliary bands and a steady region covering the remaining fluid domain. The size of the unsteady region appears to be comparable to the metachronal wavelength of the ciliary band. A systematic investigation is performed of trajectories of infinitely small (fluid) particles in the simulated unsteady ciliary driven flow. A fraction of particles appear to follow trajectories, that resemble experimentally observed particle capture events in the downstream feeding system of the polycheate Sabella penicillus, indicating that particles can be captured by ciliary systems without mechanical contact between particle and cilia. A local capture efficiency is defined and its value computed for various values of beat frequencies and other parameters. The results indicate that the simulated particle capture process is most effective when the flow field oscillates within timescales comparable to transit timescales of suspended particles passing the unsteady region near the ciliary bands. However, the computed retention efficiencies are found to be much lower than those obtained experimentally.