We investigate the uptake of a passive solute through the walls of a pulsating, fluid-filled channel into an adjacent medium in which the solute diffuses and is consumed at a constant rate. One end of the channel is open to well-mixed fluid containing the solute. The channel walls oscillate periodically in time and this prescribed motion generates steady streaming within the channel. We determine how this flow enhances the overall solute consumption (i.e. the flux of solute into the channel), the solute dispersion along the channel and the quantity of solute in the adjacent medium. The solute disperses in the channel due to the interaction between advection and transverse diffusion. The time-mean solute distribution throughout the channel and the medium is determined for a wide range of parameters. The results are applied to a new surgical technique used to treat patients with severe coronary artery disease, in which narrow tubes are created within ischemic heart muscle in an attempt to reperfuse the area directly with oxygenated blood.