  We consider the Brenier-Schr{\"o}dinger problem on compact manifolds with
boundary. In the spirit of a work by Arnaudon, Cruzeiro, L{\'e}onard and
Zambrini, we study the kinetic property of regular solutions and obtain a link
to the Navier-Stokes equations with an impermeability condition. We also
enhance the class of models for which the problem admits a unique solution.
This involves a method of taking quotients by reflection groups for which we
give several examples.
