  We show some computations related to the motion by mean curvature flow of a
submanifold inside an ambient Riemannian manifold evolving by Ricci or backward
Ricci flow. Special emphasis is given to the possible generalization of
Huisken's monotonicity formula and its connection with the validity of some
Li--Yau--Hamilton differential Harnack--type inequalities in a moving
Riemannian manifold.
