In which sense is Gruson and Raynaud's [relative dévissage][1] an "extension/ generalization" of [Grothendieck's "classical" dévissage][2] 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")?




  [1]: https://en.wikipedia.org/wiki/D%C3%A9vissage#Gruson_and_Raynaud's_relative_d%C3%A9vissages
  [2]: https://en.wikipedia.org/wiki/D%C3%A9vissage#Grothendieck's_d%C3%A9vissage_theorem