Asymptotic behaviour of Castelnuovo-Mumford regularity

Let $S$ be a polynomial ring over a field. For a graded $S$-module generated in degree at most $P$, the Castelnuovo-Mumford regularity of each of (i) its $n^{\operatorname{th}}$ symmetric power, (ii) its $n^{\operatorname{th}}$ torsion-free symmetric power and (iii) the integral closure of its $n^{\operatorname{th}}$ torsion-free symmetric power is bounded above by a linear function in $n$ with leading coefficient at most $P$. For a graded ideal $I$ of $S$, the regularity of $I^{n}$ is given by a linear function of $n$ for all sufficiently large $n$. The leading coefficient of this function is identified.