  We show that the discrete approximate volume preserving mean curvature flow
in the flat torus $\mathbb{T}^N$ starting near a strictly stable critical set
$E$ of the perimeter converges in the long time to a translate of $E$
exponentially fast. As an intermediate result we establish a new quantitative
estimate of Alexandrov type for periodic strictly stable constant mean
curvature hypersurfaces. Finally, in the two dimensional case a complete
characterization of the long time behaviour of the discrete flow with arbitrary
initial sets of finite perimeter is provided.
