  Consider a symplectic map which possesses a normally hyperbolic invariant
manifold of any even dimension with transverse homoclinic channels. We develop
a topological shadowing argument to prove the existence of Arnold diffusion
along the invariant manifold, shadowing some iterations of the inner dynamics
carried by the invariant manifold and the outer dynamics induced by the stable
and unstable foliations. In doing so, we generalise an idea of Gidea and de la
Llave in [26], based on the method of correctly aligned windows and a so-called
transversality-torsion argument. Our proof permits that the dynamics on the
invariant manifold satisfy only a non-uniform twist condition, and, most
importantly for applications, that the splitting of separatrices be small in
certain directions and thus the associated drift in actions very slow;
diffusion occurs in the directions of the manifold having non-small splitting.
Furthermore we provide estimates for the diffusion time.
