This article deals with stability of continuous-time switched systems under constrained switching. Given a family of systems, possibly containing unstable dynamics, we characterize a new class of switching signals under which the switched system generated by it and the family of systems is globally asymptotically stable. Our characterization of such stabilizing switching signals involves the asymptotic frequency of switching, the asymptotic fraction of activation of the constituent systems, and the asymptotic densities of admissible transitions among them. Our techniques employ multiple Lyapunov-like functions, and extend preceding results both in scope and applicability.