This talk presents tools for studying geodesic convexity 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 the Carlen-Gangbo sphere-like submanifolds or in Brenier's models for incompressible fluids (Lavenant, Baradat-Monsaingeon), as well as new ones such as semiconvexity in the space of couplings. The arguments revolve around a simple but versatile principle, which crucially requires no knowledge of the structure or regularity of geodesics in the submanifold. In these setting, 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 work is collaboration with Prof. Daniel Lacker.