Abstract: A thorough understanding of the skyrmion motion through nanotracks is a prerequisite to realize the full potential of spintronic applications like the skyrmion racetrack memory. One of the challenges is to place the data, i.e. skyrmions, on discrete fixed positions, e.g. below a read or write head. In the domain-wall racetrack memory, one proposed solution to this problem was patterning the nanotrack with notches. Following this approach, this paper reports on the skyrmion mobility through a nanotrack with periodic notches (constrictions) made using variations in the chiral Dzyaloshinskii-Moriya interaction. We observe that such notches induce a coupling between the mobility and the skyrmion breathing mode, which manifests itself as velocity-dependent oscillations of the skyrmion diameter and plateaus in which the velocity is independent of the driving force. Despite the fact that domain walls are far more rigid objects than skyrmions, we were able to perform an analogous study and, surprisingly, found even larger plateaus of constant velocity. For both systems it is straightforward to tune the velocity at these plateaus by changing the design of the notched nanotrack geometry, e.g. by varying the distance between the notches. Therefore, the notch-induced coupling between the excited modes and the mobility could offer a strategy to stabilize the velocity against unwanted perturbations in racetrack-like applications. In the last part of the paper we focus on the low-current mobility regimes, whose very rich dynamics at nonzero temperatures are very similar to the operating principle of recently developed probabilistic logic devices. This proves that the mobility of nanomagnetic structures through a periodically modulated track is not only interesting from a fundamental point of view, but has a future in many spintronic applications.