Molecular hydrogen is a fascinating candidate for novel bosonic and fermionic superfluids. We have studied diffusion dynamics of thin films of H_{2}, HD, and D_{2} adsorbed on a glass substrate by measurements of elasticity. The elasticity shows multiple anomalies well below the bulk triple point temperature. They are attributed to three different diffusion mechanisms of admolecules and their "freezing" into a localized state: classical thermal diffusion of vacancies, quantum tunneling of vacancies, and diffusion of molecules in the uppermost surface. The surface diffusion is active down to 1 K, below which the molecules become localized. This suggests that the surface layer of hydrogen films is on the verge of quantum phase transition to a superfluid state.