The steady-state distribution of droplets in the tube Poiseuille flow of a dilute emulsion may be modeled as a balance between the droplet drift away from the walls of the tube and the shear-induced diffusion of droplets resisting this drift. At sufficiently high flow rates, this balance results in a droplet-free layer of suspending fluid near the walls of the tube. In this paper, we demonstrate that the concentration distribution of droplets in the pressure-driven flow of a dilute emulsion through a tube with any time-dependent average velocity obeys a self-similar solution, provided the thickness of the droplet-depleted region near the walls is always non-zero. The self-similar solution is used to evaluate asymmetric oscillatory pressure-driven flows as a means of separation of the dispersed phase from the suspending fluid, and it is shown that a significant net flux of the dispersed phase may result over a cycle with the correct choice of operating parameters.