  We consider a Fokker-Planck equation which is coupled to an externally given
time-dependent constraint on its first moment. This constraint introduces a
Lagrange-multiplier which renders the equation nonlocal and nonlinear.
  In this paper we exploit an interpretation of this equation as a Wasserstein
gradient flow of a free energy ${\mathcal{F}}$ on a time-constrained manifold.
First, we prove existence of solutions by passing to the limit in an explicit
Euler scheme obtained by minimizing $h
{\mathcal{F}}(\varrho)+W_2^2(\varrho^0,\varrho)$ among all $\varrho$ satisfying
the constraint for some $\varrho^0$ and time-step $h>0$.
  Second, we provide quantitative estimates for the rate of convergence to
equilibrium when the constraint converges to a constant. The proof is based on
the investigation of a suitable relative entropy with respect to minimizers of
the free energy chosen according to the constraint. The rate of convergence can
be explicitly expressed in terms of constants in suitable logarithmic Sobolev
inequalities.
