  The present paper is devoted to the investigation of absolutely continuous
curves in asymmetric metric spaces induced by Finsler structures. Firstly, for
asymmetric spaces induced by Finsler manifolds, we show that three different
kinds of absolutely continuous curves coincide when their domains are bounded
closed intervals. As an application, a universal existence and regularity
theorem for gradient flow is obtained in the Finsler setting. Secondly, we
study absolutely continuous curves in Wasserstein spaces over Finsler manifolds
and establish the Lisini structure theorem in this setting, which characterize
the nature of absolutely continuous curves in Wasserstein spaces in terms of
dynamical transference plans concentrated on absolutely continuous curves in
base Finsler manifolds. Besides, a close relation between continuity equations
and absolutely continuous curves in Wasserstein spaces is founded. Last but not
least, we also consider nonsmooth ``Finlser-like" spaces, in which case most of
the aforementioned results remain valid. Various model examples are constructed
in this paper, which point out genuine differences between the asymmetric and
symmetric settings.
