  Gradient bounds had proved to be a very efficient tool for the control of the
rate of convergence to equilibrium for parabolic evolution equations. Among the
gradient bounds methods, the celebrated Bakry-\'Emery criterion is a powerful
way prove to convergence to equilibrium with an exponential rate. To be
satisfied, this criterion requires some form of ellipticity of the diffusion
operator. In the past few years, there have been several works extending the
Bakry-\'Emery methodology to hypoelliptic operators. Inspired by these methods,
we describe a rather simple generalization of the criterion that applies to a
large class of hypoelliptic/hypocoercive operators. We are particularly
interested in convergence to equilibrium in the Wasserstein distance and obtain
several new results in that direction.
