  In this article, we prove a general and rather flexible upper bound for the
heat kernel of a weighted heat operator on a closed manifold evolving by an
intrinsic geometric flow. The proof is based on logarithmic Sobolev
inequalities and ultracontractivity estimates for the weighted operator along
the flow, a method which was previously used by Davies in the case of a
non-evolving manifold. This result directly implies Gaussian-type upper bounds
for the heat kernel under certain bounds on the evolving distance function; in
particular we find new proofs of Gaussian heat kernel bounds on manifolds
evolving by Ricci flow with bounded curvature or positive Ricci curvature. We
also obtain similar heat kernel bounds for a class of other geometric flows.
