I think I have a proof of the following elementary lemma (although I only need the case in which the two flags are "in general position", i.e., $F^d \cap G^i$ is minimal given the dimensions of the spaces):

>Let $V$ be a finite-dimensional vector space, and $F^{\bullet}$, $G^{\bullet}$ two 
decreasing partial flags on $V$.  Suppose that for every $d$, a linear map 
$$
\phi_d \in \operatorname{End}(F^d / F^{d+1})
$$
is given that respects the decreasing partial flag on $F^d / F^{d+1}$ defined by
$$\newcommand{\dpunct}{\;\text}
G^i_d = \frac{G^i \cap F^d}{G^i \cap F^{d+1}}\dpunct.
$$
Then there exists a linear map
$
\phi \colon V \to V
$
respecting both $F^{\bullet}$ and $G^{\bullet}$ such that for every $d$, the induced endomorphism 
of $F^d / F^{d+1}$ is precisely $\phi_d$.

I'm sure this is in some sense standard. My proof certainly does not feel like something that belongs in a research paper. Does anyone know a standard reference either for this fact, or for other standard facts from which this lemma follows trivially?