The growth of graphene by molecular beam epitaxy from an elemental carbon precursor is a very promising technique to overcome some of the main limitations of the chemical vapour deposition approach, such as the possibility to synthesize graphene directly on a wide variety of surfaces including semiconductors and insulators. However, while the individual steps of the chemical vapour deposition growth process have been extensively studied for several surfaces, such knowledge is still missing for the case of molecular beam epitaxy, even though it is a key ingredient to optimise its performance and effectiveness. In this work, we have performed a combined experimental and theoretical study comparing the growth rate of the molecular beam epitaxy and chemical vapour deposition processes on the prototypical Ir (111) surface. In particular, by employing high-resolution fast X-ray photoelectron spectroscopy, we were able to follow the growth of both single- and multi-layer graphene in real time, and to identify the spectroscopic fingerprints of the different C layers. Our experiments, supported by density functional theory calculations, highlight the role of the interaction between different C precursor species and the growing graphene flakes on the growth rate of graphene. These results provide an overview of the main differences between chemical vapour deposition and molecular beam epitaxy growth and thus on the main parameters which can be tuned to optimise growth conditions.