Fitting ideals and a Grassmannian construction

Question

Let $L$ be a locally free and finitely presented sheaf over a Noetherian scheme $X$ and $$ E\overset{\varphi}\to F \to L \to 0$$ a free presentation of $L$, where $E$ and $F$ have finite ranks $n$ and $m$. Moreover, let $$ \pi:G=G(n-k, E)\to X $$ be the Grassmannian bundle of $(n-k)$-hyperplanes of sections of $E$.

For any integer $k$ we can associate to $L$ two objects:

1. The ideal sheaf $\operatorname{Fitt}_k(L) \subset \mathcal{O}_X$ over $X$, defined for example here.

2. The ideal sheaf $\operatorname{Grass}_k(L) \subset \mathcal{O}_G$ over $G$, defined below.

Def: Consider the natural short exact sequence of sheaves over $G$ $$ 0\to S \to \pi^* E \to Q \to 0 $$ where $S$ and $Q$ are the universal subbundle and quotient bundle of $G$. We define $\operatorname{Grass}_k(L)$ to be the zero locus of the morphism of sheaves $$ S\longrightarrow \pi^*E \overset{\pi^*\varphi} \longrightarrow \pi^* F $$ (i.e. the minimal ideal sheaf $I\subset\mathcal{O}_G$ such that $S/I\to \pi^* F/I$ is the zero morphism).

Now it's time to ask my question.

What is the relationship between $\operatorname{Fitt}_k(L)$ and $ \operatorname{Grass}_k(L)$ ?

Notice that we can pull back $\operatorname{Fitt}_k(L)$ using $G\overset{\pi}\to X$, and we get an ideal sheaf $\pi^* \operatorname{Fitt}_k(L) \subset \mathcal{O}_G $. Maybe it's better to compare this one with $\operatorname{Grass}_k(L)$?

Of course the above sheaves are not isomorphic, as they even have different fibers. Indeed if we choose coordinates we see that, for $W\in \pi^{-1}(x)$, the fiber $$ \pi^* \operatorname{Fitt}_k(L)_W = \operatorname{Fitt}_k(L)_x $$ is generated by the $(n-k)\times (n-k)$ minors of the matrix representation of $\varphi$, while the fibers $$ \operatorname{Grass}_k(L)_W \quad\text{and}\quad \operatorname{Grass}_k(L)_{W'} $$ are in general not the same for $W,W' \in \pi^{-1}(x)$. If I understand correctly they are generated by the entries of the matrices corresponding to $\varphi_{\mid W}$ and $\varphi_{\mid W'}$.

Intuitively, the subscheme of $G$ corresponding to $\operatorname{Grass}_k(L)$ is kind a blow-up of the subscheme of $X$ corresponding to $\operatorname{Fitt}_k(L)$: if $x \in X$ is a point where the rank of $\varphi_x$ is smaller than $t$, it gets in some sense unwinded into couples $(x, W)$ where $W$ is a $t$-plane in the kernel of $\varphi_x$.

Could you help me making my intuition precise?

Any thought, comment or correction is really appreciated.