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Using the curvature-dimension inequality proved in Part~I, we look atconsequences of this inequality in terms of the interaction between thesub-Riemannian geometry and the heat semigroup $P_t$ corresponding to thesub-Laplacian. We give bounds for the gradient, entropy, a Poincar\'einequality and a Li-Yau type inequality. These results require that thegradient of $P_t f$ remains uniformly bounded whenever the gradient of $f$ isbounded and we give several sufficient conditions for this to hold.

Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations, Part II