Mechanistic models can provide an intuitive and interpretable explanation of network growth by specifying a set of generative rules. These rules can be defined by domain knowledge about real-world mechanisms governing network growth or may be designed to facilitate the appearance of certain network motifs. In the formation of real-world networks, multiple mechanisms may be simultaneously involved; it is then important to understand the relative contribution of each of these mechanisms. In this paper, we propose the use of a conditional density estimator, augmented with a graph neural network, to perform inference on a flexible mixture of network-forming mechanisms. This event-wise mixture-of-mechanisms model assigns mechanisms to each edge formation event rather than stipulating node-level mechanisms, thus allowing for an explanation of the network generation process, as well as the dynamic evolution of the network over time. We demonstrate that our approximate Bayesian approach yields valid inferences for the relative weights of the mechanisms in our model, and we utilize this method to investigate the mechanisms behind the formation of a variety of real-world networks.