  We prove that a Finsler spacetime endowed with a smooth reference measure
whose induced weighted Ricci curvature $\smash{\mathrm{Ric}_N}$ is bounded from
below by a real number $K$ in every timelike direction satisfies the timelike
curvature-dimension condition $\smash{\mathrm{TCD}_q(K,N)}$ for all $q\in
(0,1)$. A nonpositive-dimensional version ($N \le 0$) of this result is also
shown. Our discussion is based on the solvability of the Monge problem with
respect to the $q$-Lorentz-Wasserstein distance as well as the characterization
of $q$-geodesics of probability measures. One consequence of our work is the
sharp timelike Brunn-Minkowski inequality in the Lorentz-Finsler case.
