We study a canonical model of decentralized exchange for a durable good or asset, where agents are assumed to have time-varying, heterogeneous utility types. Whereas the existing literature has focused on the special case of two types, we allow agents' utility to be drawn from an arbitrary distribution. Our main contribution is methodological: we provide a solution technique that delivers a complete characterization of the equilibrium, in closed form, both in and out of the steady state. This characterization offers a richer framework for confronting data from real-world markets and reveals a number of new economic insights. In particular, we show that heterogeneity magnifies the impact of frictions on equilibrium outcomes and that this impact is more pronounced on price levels than on price dispersion and welfare.