  We consider normal velocity of smooth sets evolving by the $s-$fractional
diffusion. We prove that for small time, the normal velocity of such sets is
nearly proportional to the mean curvature of the boundary of the initial set
for $s\in [\frac{1}{2}, 1)$ while, for $s\in (0, \frac{1}{2})$, it is nearly
proportional to the fractional mean curvature of the initial set. Our results
show that the motion by (fractional) mean curvature flow can be approximated by
fractional heat diffusion and by a diffusion by means of harmonic extension of
smooth sets.
