I have two questions concerning morphism between local cohomology groups which I think are related.

Let $G$ be a reductive group with Weyl group $W$ and $B \subset G$ a Borel. Let $X=G/B$ be the flag variety, $C(w)=BwB/B \subset X$ the Schubert cell associated to $w \in W$, and $X(w)$ its closure. Additionally fix an abelian sheaf $\mathcal{F} \in \mathit{Ab}(X)$.

- Let $v,w \in W$ with $l(w)=l(v)+1$ and we abbreviate $\operatorname{codim}(C(w))$ by $c_w$. In Kempf's paper "The Grothendieck–Cousin complex of an induced representation" he mentions in a remark after Lemma 12.6 that $H^{c_w}_{C(w)}(X,\mathcal{F}) \rightarrow H^{c_v}_{C(v)}(X,\mathcal{F})$ is trivial if $C(v) \not\subset X(w)$ (equivalently $v \nleq w$) and says that it follows by trivial support related reasons. But unfortunately I don't see the concrete argumentation. So how does it follow in more detail?

- Let $v,w \in W$ with $l(w)=l(v)$ and $v\neq w$. Is it then also true that $H^i_{X(w)}(X,\mathcal{F}) \rightarrow H^i_{X(v)}(X,\mathcal{F})$ is trivial in general or at least for $i=c_w$? (Notice that the local cohomology groups vanish for $i<c_w=c_v$.)