In causal optimal transport the set of permitted transportation plans is restricted to couplings satisfying certain causality conditions, which are given by the flow of time. In Eckstein and Cheridito (2023) the authors consider instead causal restrictions given by an associated directed graph G. Classical optimal transport and causal optimal transport can then be seen as particular instances of such G-causal optimal transport problems. We study fundamental properties of G-causal optimal transport and its induced G-Wasserstein distance. A particular focus is on determining what graph structures give a G-Wasserstein distance which respects the triangle inequality, an open problem in the original paper. Moreover, we introduce a new notion of lifted G-Wasserstein distances, which are always metrics and agree with G-Wasserstein distances on measures with continuous disintegration kernels.