We investigate flow in a tube driven by axial and radial wall oscillations that are periodic in time. Furthermore, the walls undergo periodic circumferential oscillations, the amplitude of which increases with axial distance along the tube; this wall motion results in a new mechanism for the generation of axial flows within the tube. The axial steady-streaming flows that arise as a result of circumferential oscillations are opposed to those due to the axial and radial pulsations, and are potentially as significant. We then consider the uptake of a passive solute through the walls of such a tube into an adjacent medium in which the solute diffuses and is consumed at a constant rate. The tube ends are open to well-mixed fluid containing the solute. The effect of the streaming flows on the solute flux into the tube is determined. The solute disperses along the tube due to the interaction between advection and transverse diffusion, and the time-mean solute distribution throughout the tube is determined for a wide range of parameters.