  We investigate the existence of Arnold diffusion-type orbits for systems
obtained by iterating in any order the time-one maps of a family of Tonelli
Hamiltonians. Such systems are known as 'polysystems' or 'iterated function
systems'. When specialized to families of twist maps on the cylinder, our
results are similar to those obtained by Moeckel [20] and Le Calvez [15]. Our
approach is based on weak KAM theory and is close to the one used by Bernard in
[3] to study the case of a single Tonelli Hamiltonian.
