  Let $M$ be a differentiable manifold endowed with a family of complete
Riemannian metrics $g(t)$ evolving under a geometric flow over the time
interval $[0,T[$. In this article, we give a probabilistic representation for
the derivative of the corresponding conjugate semigroup on $M$ which is
generated by a Schr\"{o}dinger type operator. With the help of this derivative
formula, we derive fundamental Harnack type inequalities in the setting of
evolving Riemannian manifolds. In particular, we establish a dimension-free
Harnack inequality and show how it can be used to achieve heat kernel upper
bounds in the setting of moving metrics. Moreover, by means of the
supercontractivity of the conjugate semigroup, we obtain a family of canonical
log-Sobolev inequalities. We discuss and apply these results both in the case
of the so-called modified Ricci flow and in the case of general geometric
flows.
