We propose a nonlinear difference-in-differences method to estimate multivariate counterfactual distributions in classical treatment and control study designs with observational data. Our approach sheds a new light on existing approaches like the changes-in-changes and the classical semiparametric difference-in-differences estimator and generalizes them to settings with multivariate heterogeneity in the outcomes. The main benefit of this extension is that it allows for arbitrary dependence and heterogeneity in the joint outcomes. We demonstrate its utility both on synthetic and real data. In particular, we revisit the classical Card \& Krueger dataset, examining the effect of a minimum wage increase on employment in fast food restaurants; a reanalysis with our method reveals that restaurants tend to substitute full-time with part-time labor after a minimum wage increase at a faster pace. A previous version of this work was entitled "An optimal transport approach to causal inference.