The chemical vapour deposition (CVD) of graphene on three polycrystalline transition metal catalysts, Co, Ni and Cu, is systematically compared and a first-order growth model is proposed which can serve as a reference to optimize graphene growth on any elemental or alloy catalyst system. Simple thermodynamic considerations of carbon solubility are insufficient to capture even basic growth behaviour on these most commonly used catalyst materials, and it is shown that kinetic aspects such as carbon permeation have to be taken into account. Key CVD process parameters are discussed in this context and the results are anticipated to be highly useful for the design of future strategies for integrated graphene manufacture.