We construct examples of Gevrey non-analytic perturbations of an integrable Hamiltonian system which give rise to an open set of unstable orbits and to a special kind of symbolic dynamics. We find an open ball in the phase space, which is transported by the Hamiltonian flow from $-\infty$ to $+\infty$ along one coordinate axis, at a speed that we estimate with respect to the size of the perturbation. Taking advantage of the hyperbolic features of this unstable system, particularly the splitting of invariant manifolds, we can also embed a random walk along this axis into the near-integrable dynamics.