  This article studies the dynamics of the strong solution of a SDE driven by a
discontinuous L\'evy process taking values in a smooth foliated manifold with
compact leaves. It is assumed that it is \textit{foliated} in the sense that
its trajectories stay on the leaf of their initial value for all times a.s.. %
Such a system is called a \textit{foliated L\'evy diffusion}. Under a generic
ergodicity assumption for each leaf, % and a continuous variation of the
ergodic measures among each others, we determine the effective behaviour of the
system subject to a small smooth perturbation of order $\e>0$, which acts
transversal to the leaves. The main result states that, on average, the
transversal component of the perturbed SDE converges uniformly to the solution
of a deterministic ODE as $\e$ tends to zero. This transversal ODE is generated
by the average of the perturbing vector field with respect to the invariant
measures of the unperturbed system and varies with the transversal height of
the leaves. We give upper bounds for the rates of convergence % The right-hand
side of this ODE is given as the average of the perturbing vector field % with
respect to the unique invariant measures of the unperturbed system on the
leaves. % We give upper bounds for the rates of convergence. and illustrate
these results for the random rotations on the circle. This article %which are
proved for pure jump L\'evy processes complements the results by Gargate and
Ruffino for SDEs of Stratonovich type to general L\'evy driven SDEs of Marcus
type.

Probability;Dynamical Systems