In modern scientific applications, observations often take the form of probability distributions. We study regression problems with distribution-valued responses and mixed distributional and Euclidean predictors. Under quadratic optimal transport, the proposed Kantorovich regression model approximates the response Kantorovich potential from the Wasserstein barycenter as a linear combination of transformed predictor Kantorovich potentials, where the transformations are determined by one-dimensional functional parameters. The formulation naturally accommodates mixed predictors, allowing Euclidean covariates to enter as scaling coefficients, and extends to multivariate distributions. We characterize functional parameter classes ensuring the intrinsic structure of the model, and establish asymptotic consistency of model parameters through the convex Sobolev loss. Real data applications include a mixed-predictor analysis of housing price distributions and an analysis of two-dimensional temperature distributions, demonstrating the flexibility and interpretability of the proposed framework.

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