Transportation of measure underlies many contemporary methods in uncertainty quantification and machine learning. Sampling can be done efficiently given an appropriate measure-transport map. We ask: what is the effect of using approximate maps in such algorithms? We proposed a new framework to analyze the approximation power of measure transport. This framework applies to existing algorithms, but also suggests new ones. At the core of our analysis is the theory of optimal transport regularity, approximation theory, and an emerging class of inequalities, previously studied in the context of surrogate models and density estimation.