  Dynamics of topological magnetic textures are typically induced externally
by, e.g.~magnetic fields or spin/charge currents. Here, we demonstrate the
effect of the internal-to-the-system geometry-induced motion of a domain wall
in a curved nanostripe. Being driven by the gradient of the curvature of a
biaxial stripe, transversal domain walls acquire remarkably high velocities of
up to 100 m/s and do not exhibit any Walker-type speed limit. We pinpoint that
the inhomogeneous distribution of the curvature-induced Dzyaloshinskii--Moriya
interaction is a driving force for the motion of a domain wall. Although we
showcase our approach on the specific Euler spiral geometry, the approach is
general and can be applied to a wide class of geometries.
