Asymptotic depth of twisted higher direct image sheaves

Let $\pi:X \rightarrow X_{0}$ be a projective morphism of schemes, such that $X_{0}$ is Noetherian and essentially of finite type over a field $K$. Let $i \in \mathbb{N}_{0}$, let ${\mathcal{F}}$ be a coherent sheaf of ${\mathcal{O}}_{X}$-modules and let ${\mathcal{L}}$ be an ample invertible sheaf over $X$. Let $Z_{0} \subseteq X_{0}$ be a closed set. We show that the depth of the higher direct image sheaf ${\mathcal{R}}^{i}\pi_{*}({\mathcal{L}}^{n} \otimes_{{\mathcal{O}}_{X}} {\mathcal{F}})$ along $Z_{0}$ ultimately becomes constant as $n$ tends to $-\infty$, provided $X_{0}$ has dimension $\leq 2$. There are various examples which show that the mentioned asymptotic stability may fail if $\dim(X_{0}) \geq 3$. To prove our stability result, we show that for a finitely generated graded module $M$ over a homogeneous Noetherian ring $R=\bigoplus_{n \geq 0}R_{n}$ for which $R_{0}$ is essentially of finite type over a field and an ideal $\mathfrak{a}_{0} \subseteq R_{0}$, the $\mathfrak{a}_{0}$-depth of the $n$-th graded component $H^{i}_{R_{+}}(M)_{n}$ of the $i$-th local cohomology module of $M$ with respect to $R_{+}:=\bigoplus_{k>0}R_{k}$ ultimately becomes constant in codimension $\leq 2$ as $n$ tends to $-\infty$.