A Bakry-\'Emery approach to Lipschitz transportation on manifolds

On weighted Riemannian manifolds we prove the existence of globally Lipschitz transport maps between the weight (probability) measure and log-Lipschitz perturbations of it, via Kim and Milman's diffusion transport map, assuming that the curvature-dimension condition $\mathrm{CD}(\rho_{1}, \infty)$ holds, as well as a second order version of it, namely $\Gamma_{3} \geq \rho_{2} \Gamma_{2}$. We get new results as corollaries to this result, as the preservation of Poincar\'e's inequality for the exponential measure on $(0,+\infty)$ when perturbed by a log-Lipschitz potential and a new growth estimate for the Monge map pushing forward the gamma distribution on $(0,+\infty)$ (then getting as a particular case the exponential one), via Laguerre's generator.