The interplay between thermal relaxation and statistical dissociation in binary Morse clusters (AB)N has been investigated using numerical simulations and simple statistical approaches, for a variety of interaction parameters covering miscible and non-miscible regimes. While all clusters exhibit a core/shell phase separation pattern in their most stable, T = 0 structure, different melting mechanisms are identified depending on the ranges and their mismatch, including two-step melting of the surface and the core or premelting as alloying. The preference for emitting A or B particles upon evaporation has been evaluated assuming that the cluster is either thermally equilibrated or vibrationally excited in its ground state structure, and compared to the predictions of the Weisskopf theory. The variations of the dissociation rate constants with increasing energy and the branching ratio between the two channels show significant differences in both cases, especially when the clusters are miscible and bound by short-range forces, which indicates that the time scale for evaporation is much shorter than the equilibration time. Our results suggest that dissociation properties could be used to test the ergodic hypothesis in such compounds.