Geometric mechanics models of locomotion have provided insight into how robots and animals use environmental interactions to convert internal shape changes into displacement through the world, encoding this relationship in a ``motility map''. A key class of such motility maps arises from (possibly anisotropic) linear drag acting on the system's individual body parts, formally described via Riemannian metrics on the motions of the system's individual body parts. The motility map can then be generated by invoking a sub-Riemannian constraint on the aggregate system motion under which the position velocity induced by a given shape velocity is that which minimizes the power dissipated via friction. The locomotion of such systems is ``geometric'' in the sense that the final position reached by the system depends only on the sequence of shapes that the system passes through, but not on the rate with which the shape changes are made. In this paper, we consider a far more general class of systems in which the drag may be not only anisotropic (with different coefficients for forward/backward and left/right motions), but also asymmetric (with different coefficients for forward and backward motions). Formally, including asymmetry in the friction replaces the Riemannian metrics on the body parts with Finsler metrics. We demonstrate that the sub-Riemannian approach to constructing the system motility map extends naturally to a sub-Finslerian approach and identify system properties analogous to the constraint curvature of sub-Riemannian systems that allow for the characterization of the system motion capabilities.