Let {x∈(·)} be a sequence of solutions to an ordinary differential equation with random right sides (due to input noise {ξ∈(·)}) and which converges weakly to a diffusionx(·) with unique invariant measure µ(.). Let µ(t,·) denote the measure ofx(t), and suppose that µ(t,·)-?µ(·) weakly. The paper shows, under reasonable conditions, that the measures ofx(t) are close to µ(·) for largetand small ∈. In applications, such information is often more useful than the simple fact of the weak convergence. The noise ξ∈(·) need not be bounded, the pair (x∈(·), ξ(·)) need not be Markovian (except for the unbounded noise case), and the dynamical terms need not be smooth. The discrete parameter case is also treated, and several examples arising in control and communication theory are given.