  The defining equation $(\ast):\ \dot \omega\_t=-F'(\omega\_t),$ of a gradient
flow is kinetic in essence. This article explores some dynamical (rather than
kinetic) features of gradient flows (i) by embedding equation $(\ast)$ into the
family of slowed down gradient flow equations: $\dot \omega ^{
\varepsilon}\_t=- \varepsilon F'( \omega ^{ \varepsilon}\_t),$ where
$\varepsilon>0$, and (ii) by considering the \emph{accelerations} $\ddot \omega
^{ \varepsilon}\_t$. We shall focus on Wasserstein gradient flows. Our approach
is mainly heuristic. It relies on Otto calculus.A special formulation of the
Schr{\"o}dinger problem consists in minimizing some action on the Wasserstein
space of probability measures on a Riemannian manifold subject to fixed initial
and final data. We extend this action minimization problem by replacing the
usual entropy, underlying Schr{\"o}dinger problem, with a general function of
the Wasserstein space. The corresponding minimal cost approaches the squared
Wasserstein distance when some fluctuation parameter tends to zero. We show
heuristically that the solutions satisfy a Newton equation, extending a recent
result of Conforti. The connection with Wasserstein gradient flows is
established and various inequalities, including evolutional variational
inequalities and contraction inequality under curvature-dimension condition,
are derived with a heuristic point of view. As a rigorous result we prove a new
and general contraction inequality for the Schr{\"o}dinger problem under a
Ricci lower bound on a smooth and compact Riemannian manifold.
