Integration against an intractable probability measure is among the fundamental challenges of statistical inference, particularly in the Bayesian setting. A principled approach to this problem seeks a deterministic coupling of the measure of interest with a tractable “reference” measure (e.g., a standard Gaussian). This coupling is induced by a transport map, and enables direct simulation from the desired measure simply by evaluating the transport map at samples from the reference. Yet characterizing such a map—e.g., representing and evaluating it—grows challenging in high dimensions. The central contribution of this paper is to establish a link between the Markov properties of the target measure and the existence of certain low-dimensional couplings, induced by transport maps that are sparse or decomposable. Our analysis not only facilitates the construction of couplings in high-dimensional settings, but also suggests new inference methodologies. For instance, in the context of nonlinear and non-Gaussian state space models, we describe new online and single-pass variational algorithms that characterize the full posterior distribution of the sequential inference problem using operations only slightly more complex than regular filtering.