  In this article, we give a trajectorial proof of a kinetic Poincar\'e
inequality which plays an important role in the De Giorgi-Nash-Moser theory for
kinetic equations. The present work improves a result due to J. Guerand and C.
Mouhot [10] in several directions. We use kinetic trajectories along the vector
fields $\partial_t + v \cdot \nabla_x$ and $\partial_{v_i}$, $i = 1,\dots, d$
and do not rely on higher-order commutators such as $[\partial_{v_i},\partial_t
+ v \cdot \nabla_x] = \partial_{x_i}$ or on the fundamental solution. The
presented method also applies to more general hypoelliptic equations. We
illustrate this by studying a Kolmogorov equation with $k$ steps.
