  Multiple-$Q$ spin textures, such as magnetic bubble and skyrmion lattices,
can be driven into motion by external stimuli. The motion of spin textures
affects the electronic states. Here we show that to describe correctly the
electronic dynamics, the momentum space needs to be transcended to higher
dimensions by including the ancillary dimensions associated with phason modes
of the translational motion of the spin textures. The electronic states have
non-trivial topology characterized by the first and second Chern numbers in the
high dimensional hybrid momentum space. This gives rise to an anomalous
electric charge transport due to the motion of spin textures. By deforming the
spin textures, a nonlinear response associated with the second Chern number can
be induced. The charge transport is derived from the semi-classical equation of
motion of electrons that depends on the Berry curvature in the hybrid momentum
space. Our results suggest that the motion of multiple-$Q$ spin textures has
significant effects on the electronic dynamics and provides a new platform to
explore high dimensional topological physics.
