The long time behavior of the dynamics of an example of a fast-slow system of ordinary differential equations is examined. The system is derived from a spatial discretization of a Korteweg-de Vries-Burgers type equation, with fast dispersion and slow diffusion. The discretization is based on a model developed by Goodman and Lax, that is composed of a fast system drifted by a slow forcing term. A difficulty to invoke available multiscale methods arises since the underlying system does not possess a natural split to fast and slow state variables. Our approach depicts the limit behavior as a Young measure with values being invariant measures of the fast contribution to the flow. The slow contribution to the dynamics causes these invariant measures to drift. We keep track of this drift via slowly evolving observables. Averaging equations for the latter lead to computation of characteristic features of the motion and the location the invariant measures. Such computations are presented in the