We study the diffusion dynamics, the diffusion mechanisms, and the adsorption energetics of Ag, Au, Cu, and Pd dimers, as well as of Ag trimers on single-layer graphene (SLG) by means of ab initio molecular dynamics (AIMD) simulations and density-functional theory (DFT) calculations. The simulations show that Ag, Cu, and Au clusters exhibit a super-diffusive pattern characterized by long jumps, which can be explained by the flat potential energy landscape (PEL) (corrugation of a few tens of meV) encountered by those clusters on SLG. Pd dimers, instead, diffuse in a pattern that is reminiscent of conventional random walk, which is consistent with a significantly rougher PEL of the order of 100 meV. Moreover, our data show that all clusters exhibit diffusion mechanisms that include both concerted translation and rotation. The overall results of the present study provide key insights for modeling the growth of metal layers and nanostructures on graphene and other van der Waals materials, which is a prerequisite for the directed growth of multifunctional metal contacts in a broad range of enabling devices.