The diffusion processes and kinetic barriers of individual carbon adatoms and clusters on graphene surfaces are investigated to provide fundamental understanding of the physics governing epitaxial growth of multilayer graphene. It is found that individual carbon adatoms form bonds with the underlying graphene whereas the interaction between graphene and carbon clusters, consisting of 6 atoms or more, is very weak being van der Waals in nature. Therefore, small carbon clusters are quite mobile on the graphene surfaces and the diffusion barrier is negligibly small (∼6 meV). This suggests the feasibility of high-quality graphene epitaxial growth at very low growth temperatures with small carbon clusters (e.g., hexagons) as carbon source. We propose that the growth mode is totally different from 3-dimensional bulk materials with the surface mobility of carbon hexagons being the highest over graphene surfaces that gradually decreases with further increase in cluster size.