  We consider a planar geometric flow in which the normal velocity is a
nonlocal variant of the curvature. The flow is not scaling invariant and in
fact has different behaviors at different spatial scales, thus producing
phenomena that are different with respect to both the classical mean curvature
flow and the fractional mean curvature flow.
  In particular, we give examples of neckpinch singularity formation, and we
discuss convexity properties of the evolution.
  We also take into account traveling waves for this geometric flow, showing
that a new family of $C^{1,1}$ and convex traveling sets arises in this
setting.
