Small deterministic and random perturbations of dynamical systems or stochastic processes will be considered. If the original system has a unique stationary regime (stationary distribution) then, under some natural additional assumptions, the perturbed system will be close, in a sense, to the original one even on long time intervals. But if the non-perturbed system has multiple invariant measures, the perturbations, in an appropriate time scale, may lead to a drift in the cone of invariant measures. This drift determines the long-term evolution of the perturbed system. I will show how this general idea works in concrete problems: Perturbations of systems with a finite number of "asymptotically stable" invariant distributions, of oscillators, of Landau-Lifshitz magnetization equation and its generalizations will be considered. The Neumann problem for second order elliptic equations with a small parameter will be also examined.