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