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Recent experiments have shown that rapid kinematic mixing occurs in the pulmonary alveoli. Here the Reynolds number is very small, there is recirculation in the alveolar cavity and the alveolar walls move periodically. We have recently shown that non-diffusing particles move chaotically in a two-dimensional model flow with the above features. In parts of the lung, however, there is asynchrony between the wall motion and the ductal flow immediately outside the alveolus. The extent to which this asynchrony affects kinematic mixing in real alveoli is not yet known. The purpose of this paper is to describe the effect of asynchrony on chaotic advection in our two-dimensional model, in order to understand the circumstances in which this becomes significant.