We study geodesic convexity properties of various functionals on submanifolds of Wasserstein spaces with their induced geometry. We obtain short new proofs of several known results, such as the strong convexity of entropy on sphere-like submanifolds due to Carlen-Gangbo, as well as new ones, such as the $\lambda$-convexity of entropy on the space of couplings of $\lambda$-log-concave marginals. The arguments revolve around a simple but versatile principle, which crucially requires no knowledge of the structure or regularity of geodesics in the submanifold (and which is valid in a general metric spaces): If the EVI($\lambda$) gradient flow of a functional exists and leaves a submanifold invariant, then the restriction of the functional to the submanifold is geodesically $\lambda$-convex. In these settings, we derive strengthened forms of Talagrand (and HWI inequalities) on submanifolds, which we show to be related to large deviation bounds for conditioned empirical measures. This is joint work with Louis-Pierre Chaintron.