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