It is investigated to what extent the trajectories of a stochastically switched (blinking) system follow the corresponding trajectories of the averaged system. Four cases have to be distinguished, depending on whether or not the averaged system has a unique attractor and whether or not the attractor(s) is (are) invariant under the dynamics of the blinking system. The corresponding asymptotic behavior of the trajectories of the blinking system is described and illustrative examples are given