In this paper, we present a novel interpretation of the Weisfeiler-Lehman (WL) distance introduced by \cite{chen2022weisfeilerlehman} using concepts from stochastic processes. The WL distance aims compares graphs with node features, has the same discriminative power as the classic Weisfeiler-Lehman graph isomorphism test and has deep connections to the Gromov-Wasserstein distance. Our interpretation connects the WL distance to the literature on distances for stochastic processes, which also makes the interpretation of the distance more accessible and intuitive. We further explore the connections between the WL distance and certain Message Passing Neural Networks, and discuss the implications of the WL distance for understanding the Lipschitz property and the universal approximation results for these networks.