  We estimate the heat kernel on a closed Riemannian manifold $M$, with
$dim(M)\geq 3$, evolving under the Ricci-harmonic map flow and the result
depends on some constants arising from a Sobolev imbedding theorem. In a
special case, when the scalar curvature satisfies a certain natural inequality,
we obtain, as a corollary, a bound similar to the one known for the fixed
metric case.
