Near-Lipschitz stability of the Kim--Milman flow map

We prove that the Kim--Milman flow map enjoys favorable stability properties with respect to variations in the target measure, provided that one of the target measures is sufficiently regular. Our results include stability in relative entropy, and more notably, Lipschitz stability in the $2$-Wasserstein distance up to a logarithmic factor. We complement our results with a general existence theorem for these maps for any target measure with finite second moment.