By M. Struwe [On the evolution of harmonic maps in higher dimension, J. Differ. Geom., to appear] and G. Huisken [J. Differ. Geom. 31, No. 1, 285-299 (1990; Zbl 0694.53005)] monotonicity formulas have been proved for harmonic maps heat flow on Euclidean domain and for the mean curvature flow of a hypersurface in Euclidean space. In this paper these results are geeralized to the case of flows on a compact manifold and to Yang-Mills heat flow. The proof uses as essential tool a matrix Harnack inequality given previously by the author [Commun. Anal. Geom. 1, 113-126 (1993)].