  This paper studies the asymptotic behaviour of the solution of a differential
equation perturbed by a fast flow preserving an infinite measure. This question
is related with limit theorems for non-stationary Birkhoff integrals. We
distinguish two settings with different behaviour: the integrable setting (no
averaging phenomenon) and the case of an additive "centered" perturbation term
(averaging phenomenon). The paper is motivated by the case where the
perturbation comes from the Z-periodic Lorentz gas flow or from the geodesic
flow over a Z-cover of a negatively curved compact surface. We establish limit
theorems in more general contexts.
