There exist infinitely many ways to transport one probability distribution to another. For a given source measure, a property of fundamental interest for theoreticians, statistician, and practitioners is the stability of a family of transport maps: how much do the maps change as a function of the target measure? In this talk, we will discuss tight stability properties for two transport maps which arise in the computational literature: the entropic transport map and the Kim--Milman flow map. As by-products of our analysis, we prove novel stability bounds for optimal transport maps in restrictive settings, and new transportation-information inequalities between unbounded measures. This is joint work with Vincent Divol (ENSAE), Jon Niles-Weed (New York University), and Sinho Chewi (Yale University), Matthew S. Zhang (UToronto)