Endowing the configuration space of a robot with an appropriate metric structure and characterizing and computing the corresponding geodesics are central issues in motion planning. As recently observed in de Mont-Marin et al. (A minimum swept-volume metric structure for configuration space. In: IEEE international conference on robotics and automation (ICRA), pp 3686–3692, 2023), the geodesics of \(\mathrm {SE}(2)\) equipped with the so called minimum swept-volume distance exhibit in practice a behavior akin to the turnpike property in optimal control, with transient phases separated by a longer steady state close to prototypical trajectories, the turnpikes (Trélat and Zuazua (2015) J Differ Equ 258(1):81–114). This presentation gives a theoretical counterpoint to this empirical observation with a formal definition of geodesic turnpikes using vector fields on Finsler manifolds, a simple differential characterization of geodesics in the case where the manifold is a Lie group and the Finsler distance is left-invariant, and, in the case where the corresponding operator is also reversible, a conjecture characterizing the turnpikes by vector fields satisfying simple conditions in the corresponding Lie algebras. As a proof of concept, closed-form (resp. numerical) procedures for computing the vector fields predicted by this conjecture are given for \(\mathrm {SE}(2)\) equipped with the left-invariant Riemannian (resp. minimum swept-volume) distance introduced in Žefran et al. (IEEE Trans Robot Autom 14(4):576–589, 1998) (resp. de Mont-Marin et al. (A minimum swept-volume metric structure for configuration space. In: IEEE international conference on robotics and automation (ICRA), pp 3686–3692, 2023)) for rectangular shapes. The solutions empirically match, in both cases, the observed turnpike behavior of the corresponding geodesics. In the minimum swept-volume distance case, using the turnpikes for initialization also yields an order of magnitude speedup in computing geodesics.