Stability analysis is developed for nonlinear nonautonomous switched systems, trajectories of which admit, generally speaking, a nonunique continuation on the right. For these systems Krasovskii-LaSalle's invariance principle is extended in such a manner to remain true even in the nonautonomous case. In addition to a nonsmooth Lyapunov function with negative-semidefinite time derivatives along the system trajectories, the extended invariance principle involves a coupled indefinite function to guarantee asymptotic stability of the system in question. As an illustration of the capabilities of this principle, a switched regulator of a fully-actuated manipulator with frictional joints is constructed.