The need to reason about uncertainty in large, complex, and multimodal data sets has become increasingly common across modern scientific environments. The ability to transform samples from one distribution $P$ to another distribution $Q$ enables the solution to many problems in machine learning (e.g., Bayesian inference, generative modeling) and has been actively pursued from theoretical, computational, and application perspectives across the fields of information theory, computer science, and biology. Performing such transformations in general still leads to computational difficulties, especially in high dimensions. Here, we consider the problem of computing such “measure transport maps” with efficient and parallelizable methods. Under the mild assumptions that $P$ need not be known but can be sampled from and that the density of $Q$ is known up to a proportionality constant, and that $Q$ is log-concave, we provide in this work a convex optimization problem pertaining to relative entropy minimization. We show how an empirical minimization formulation and polynomial chaos map parameterization can allow for learning a transport map between $P$ and $Q$ with distributed and scalable methods. We also leverage findings from nonequilibrium thermodynamics to represent the transport map as a composition of simpler maps, each of which is learned sequentially with a transport cost regularized version of the aforementioned problem formulation. We provide examples of our framework within the context of Bayesian inference for the Boston housing data set and generative modeling for handwritten digit images from the MNIST data set.