We consider systems of weakly interacting jump processes on heterogeneous random graphs and their large population limit. The interaction is of mean field type weighted by the underlying graphon. A law of large numbers result is established as the system size increases and the underlying graphons converge. The limit is given by a graphon particle system consisting of independent but heterogeneous nonlinear Markovian processes whose probability distributions are fully coupled. Individual-based epidemic models, as an application, and Join-the-shortest-queue(d) systems, as a relevant queueing model, will be briefly discussed.