In which sense is Gruson and Raynaud's relative dévissage an "extension/ generalization" of Grothendieck's "classical" dévissage except that the first one works in "relative" setting (ie with a morphism $f:X \to S$), the later not? Are there more subtle connections?

The point is that except this I not see any deeper intertwinings between these two dévissage legitimating to call the former to be extension of the later. Seemingly Grothendieck's goal was to find relatively weak set of sufficient conditions for a subcategory $\mathcal{C} \subset (\text{Coh} \mathcal{O}_X-\text{Mod})$ one can "check by hand" such that $\mathcal{C}$ is already the whole $(\mathcal{O}_X-\text{Mod})$.

So at all Grothendieck's dévissage is a concrete criterion statement on $\mathcal{C}$, and following the original proof one sees that the strategy consists of a cascade of reduction steps on an arbitrary coherent sheaf $\mathcal{F}$ on $X$ weakening/ literally "unscrewing" step by step (think that's the reason for the name Grothendieck gave to this principle) the assumptions on it leading to conclusion that it is contained in $\mathcal{C}$.

On the other hand, Gruson and Raynaud's relative dévissage is roughly speaking an additional datum a pointed morphism $f:(X,x) \to (S,s)$ can carry with it or not, having at first glance more a flavour of a descent datum. But it not really involves a criterion or technique, so far I see.

Seemingly the advantage for imposing/using such additional datum is that it sometimes allows to use some induction arguments on the length $r$ of such a concatenation of relative devissage data indexed by relative dimension vector $(n_1,..., n_r)$. But seemingly this data not caries an immediate relation to Grothendieck's devissage formulated as a criterion (at least I not still not see it).

Now the question is if there is a deeper connection between Grothendieck's and Gruson & Raynaud's devissage approaches except that just one is declared for a fixed scheme ("absolute" setting), the other for a morphism ("relative setting")?

In which sense is Gruson and Raynaud's relative dévissage an "extension/ generalization" of Grothendieck's "classical" dévissage concept except that the first one works in "relative" setting (ie with a morphism $f:X \to S$), the later not? Are there more subtle connections?

The point is that except this I not see any deeper intertwinings between these two dévissage legitimating to call the former to be extension of the later. Seemingly Grothendieck's goal was to find relatively weak set of sufficient conditions for a subcategory $\mathcal{C} \subset (\text{Coh} \mathcal{O}_X-\text{Mod})$ one can "check by hand" such that $\mathcal{C}$ is already the whole $(\mathcal{O}_X-\text{Mod})$.

So at all Grothendieck's dévissage is a concrete criterion statement on $\mathcal{C}$, and following the original proof one sees that the strategy consists of a cascade of reduction steps on an arbitrary coherent sheaf $\mathcal{F}$ on $X$ weakening/ literally "unscrewing" step by step (think that's the reason for the name Grothendieck gave to this principle) the assumptions on it leading to conclusion that it is contained in $\mathcal{C}$.

On the other hand, Gruson and Raynaud's relative dévissage is roughly speaking an additional datum to a given pointed morphism $f:(X,x) \to (S,s)$ it can carry with it or not, having at first glance more a flavour of a descent datum. But it not really (at least I not see it yet) involves a criterion or gives rise technique like in case Grothendieck's version, so far I see.

Seemingly the advantage for imposing/using such additional datum is that it sometimes allows to use some induction arguments on the length $r$ of such a concatenation of relative devissage data indexed by relative dimension vector $(n_1,..., n_r)$. But seemingly this data not caries an immediate relation to Grothendieck's devissage formulated as a criterion (at least I not still not see it).

Now the question is if there is a deeper connection between Grothendieck's and Gruson & Raynaud's devissage approaches except that just one is declared for a fixed scheme ("absolute" setting), the other for a morphism ("relative setting")?

In which sense is Gruson and Raynaud's relative dévissage an "extension/ generalization" of Grothendieck's "classical" dévissage except that the first one works in "relative" setting (ie with a morphism $f:X \to S$), the latter not?

The point is that except this I not see any deeper intertwinings between these two dévissage legitimating to call the former to be extension of the latter. Seemingly Grothendieck's goal was to find relatively weak set of sufficient conditions for a subcategory $\mathcal{C} \subset (\text{Coh} \mathcal{O}_X-\text{Mod})$ one can "check by hand" such that $\mathcal{C}$ is already the whole $(\mathcal{O}_X-\text{Mod})$. So at all Grothendieck's dévissage is a concrete criterion statement on $\mathcal{C}$,

while Gruson and Raynaud's relative dévissage is roughly speaking an additional data a pointed morphism $f:(X,x) \to (S,s)$ can carry with it or not, having at first glance more a flavour of a descent datum.

Seemingly the advantage for imposing/using such additional datum is that it sometimes allows some induction arguments on the length $r$ of such a concatenation of relative devissage data indexed by relative dimension vector $(n_1,..., n_r)$. But seemingly this data not caries an immediate relation to Grothendieck's devissage formulated as a criterion, or not?

Now the question is if there is a deeper connection between Grothendieck's and Gruson & Raynaud's devissage approaches except that just one is declared for a fixed scheme ("absolute" setting), the other for a morphism ("relative setting")?

In which sense is Gruson and Raynaud's relative dévissage an "extension/ generalization" of Grothendieck's "classical" dévissage except that the first one works in "relative" setting (ie with a morphism $f:X \to S$), the later not? Are there more subtle connections?

The point is that except this I not see any deeper intertwinings between these two dévissage legitimating to call the former to be extension of the later. Seemingly Grothendieck's goal was to find relatively weak set of sufficient conditions for a subcategory $\mathcal{C} \subset (\text{Coh} \mathcal{O}_X-\text{Mod})$ one can "check by hand" such that $\mathcal{C}$ is already the whole $(\mathcal{O}_X-\text{Mod})$. 

So at all Grothendieck's dévissage is a concrete criterion statement on $\mathcal{C}$, and following the original proof one sees that the strategy consists of a cascade of reduction steps on an arbitrary coherent sheaf $\mathcal{F}$ on $X$ weakening/ literally "unscrewing" step by step (think that's the reason for the name Grothendieck gave to this principle) the assumptions on it leading to conclusion that it is contained in $\mathcal{C}$.

On the other hand, Gruson and Raynaud's relative dévissage is roughly speaking an additional datum a pointed morphism $f:(X,x) \to (S,s)$ can carry with it or not, having at first glance more a flavour of a descent datum. But it not really involves a criterion or technique, so far I see.

Seemingly the advantage for imposing/using such additional datum is that it sometimes allows to use some induction arguments on the length $r$ of such a concatenation of relative devissage data indexed by relative dimension vector $(n_1,..., n_r)$. But seemingly this data not caries an immediate relation to Grothendieck's devissage formulated as a criterion (at least I not still not see it).

Now the question is if there is a deeper connection between Grothendieck's and Gruson & Raynaud's devissage approaches except that just one is declared for a fixed scheme ("absolute" setting), the other for a morphism ("relative setting")?