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In the current paper,under the transverse Ricci flow on a totally geodesicRiemannian foliation, we prove two types of differential Harnack inequalities(Li-Yau gradient estimate) for the positive solutions of the heat equationassociated with the time dependent horizontal Laplacian operators. We also geta time dependent version of the generalized curvature dimension inequality. Asa consequence of aforementioned results, we also get parabolic Harnackinequalities and heat kernel upper bounds.

Harnack inequalities on totally geodesic foliations with transverse Ricci flow