In this paper, we apply a total variation approach to bridge the stability criteria for nonlinear time-varying systems and nonlinear switched systems. In particular, we derive a set of stability conditions applying to both nonlinear time-varying systems and nonlinear switched systems. We show that the derived stability conditions, when applied to nonlinear time-varying systems and nonlinear switched systems, recover the existing stability results in the literature. We also show that the derived stability conditions can be applied to qualitatively recover a unified stability criterion for slowly time-varying linear systems and switched linear systems proposed in our recent work.