  We prove that skew systems with a sufficiently expanding base have
approximate exponential decay of correlations, meaning that the exponential
rate is observed modulo an error. The fiber maps are only assumed to be
Lipschitz regular and to depend on the base in a way that guarantees diffusive
behaviour on the vertical component. The assumptions do not imply an hyperbolic
picture and one cannot rely on the spectral properties of the transfer
operators involved. The approximate nature of the result is the inevitable
price one pays for having so mild assumptions on the dynamics on the vertical
component. However, the error in the approximation goes to zero when the
expansion of the base tends to infinity. The result can be applied beyond the
original setup when combined with acceleration or conjugation arguments, as our
examples show.
