  In this paper we introduce the dynamical Schr\"odinger problem on abstract
metric spaces, defined for a wide class of entropy and Fisher information
functionals. Under very mild assumptions we prove a generic Gamma-convergence
result towards the geodesic problem as the noise parameter
$\varepsilon\downarrow 0$. We also investigate the connection with geodesic
convexity of the driving entropy, and study the dependence of the entropic cost
on the parameter $\varepsilon$. Some examples and applications are discussed.
