We study how barter exchanges should be conducted through a centralized mechanism in a dynamically evolving agent pool with time- and compatibility-based preferences. We derive the dynamically efficient two-way and multi-way exchange mechanisms that maximize total discounted exchange surplus. Recently several live-donor kidney exchange programmes were established to swap incompatible donors of end-stage kidney disease patients. Since kidney exchange can be modelled as a special instance of our more general model, dynamically efficient kidney exchange mechanisms are derived as corollaries. We make policy recommendations using simulations.