  We establish the well-posedness of the nonlocal mean curvature flow of order
${\alpha\in(0,1)}$ for periodic graphs on $\mathbb{R}^n$ in all subcritical
little H\"older spaces ${\rm h}^{1+\beta}(\mathbb{T}^n)$ with $\beta\in(0,1)$.
Furthermore, we prove that if the solution is initially sufficiently close to
its integral mean in ${\rm h}^{1+\beta}(\mathbb{T}^n)$, then it exists globally
in time and converges exponentially fast towards a constant. The proofs rely on
the reformulation of the equation as a quasilinear evolution problem, which is
shown to be of parabolic type by a direct localization approach, and on
abstract parabolic theories for such problems.
