We consider a class of graph-valued stochastic processes in which each vertex has a type that fluctuates randomly over time. Collectively, the paths of the vertex types up to a given time determine the probabilities that the edges are active or inactive at that time. Our focus is on the evolution of the associated empirical graphon in the limit as the number of vertices tends to infinity, in the setting where fluctuations in the graph-valued process are more likely to be caused by fluctuations in the vertex types than by fluctuations in the states of the edges given these types. We derive both sample-path large deviation principles and convergence of stochastic processes. We demonstrate the flexibility of our approach by treating a class of stochastic processes where the edge probabilities depend not only on the fluctuations in the vertex types but also on the state of the graph itself.