Causal optimal transport and the related adapted Wasserstein distance have recently been popularized as a more appropriate alternative to the classical Wasserstein distance in the context of stochastic analysis and mathematical finance. In this talk, we establish some interesting consequences of causality for transports between laws of continuous time stochastic processes, such as SDEs and Gaussian processes. In particular, these (bi-)causal transports admit stochastic integral representations, from which we can establish topological properties and compute explicitly the adapted Wasserstein distance between Gaussian Volterra processes. Time permitting, we will discuss the stability and approximation of the adapted Wasserstein distance to address the cases where an explicit computation is not known. This talk is based on joint works with Prof. Rama Cont and Y. Jiang.