  Let $(R,m, \kappa)$ be a local ring. We give a characterization of
$R$-modules $M$ whose local cohomology is finite length up to some index in
terms of asymptotic vanishing of Koszul cohomology on parameter ideals up to
the same index. In particular, we show that a quasi-unmixed module $M$ is
asymptotically Cohen-Macaulay if and only if $M$ is Cohen-Macaulay on the
punctured spectrum if and only if $\sup\{\ell(H^i(f_1, \ldots, f_d;M))\mid
\sqrt{f_1, \ldots, f_d} = m \mbox{, } i< d\}<\infty$.
