We consider a class of multi-degree-of-freedom Hamiltonian systems having saddle-centres at which all eigenvalues are purely imaginary except a pair of positive and negative ones, and to which there are homoclinic orbits. In our situation, there exist whiskered invariant tori near the saddle-centres. We develop a Melnikov-type technique for detecting the existence of orbits transversely homoclinic or heteroclinic to the invariant tori. We also show that the systems are nonintegrable in an appropriate meaning and Arnold diffusion type motions occur if such homoclinic or heteroclinic orbits exist. Our theory is applied to systems with potentials and a concrete example is given.