In this paper we consider switched nonlinear systems with otherwise arbitrary compact index sets and subjected to different switching constraints. We present invariance principles for these systems and derive, by using observability-like notions, some convergence and asymptotic stability criteria. These results enable us to analyze the stability of solutions of switched systems with restrictions originating from the timing of the switchings, state-dependent constrained switching, and switching whose logic has memory; i.e., the active subsystem can only switch to a prescribed subset of subsystems.