The limit behaviour of $m^kM$

Let $S$ be the polynomial ring over a field $K$ in a finite set of variables, and let $m$ be the graded maximal ideal of $S$. For a finitely generated graded $S$-module $M$ and all integers $k\gg 0$, we show that $m^kM$ is componentwise linear, we describe the pattern of the Betti-diagram of $m^kM$ when $M$ is an ideal and $char(K)=0$, and show that $m^kM$ has linear quotients if $M$ is a monomial ideal