We followed by X-ray Photoelectron Spectroscopy (XPS) the time evolution of graphene layers obtained by annealing 3C SiC(111)/Si(111) crystals at different temperatures. The intensity of the carbon signal provides a quantification of the graphene thickness as a function of the annealing time, which follows a power law with exponent 0.5. We show that a kinetic model, based on a bottom-up growth mechanism, provides a full explanation to the evolution of the graphene thickness as a function of time, allowing to calculate the effective activation energy of the process and the energy barriers, in excellent agreement with previous theoretical results. Our study provides a complete and exhaustive picture of Si diffusion into the SiC matrix, establishing the conditions for a perfect control of the graphene growth by Si sublimation.