Derivators and fibred $\infty$-categories

In his Cohomological methods in intersection theory, Cisinski writes:

"[...] note however that, by Balzin’s work [Bal19, Theorem 2], it is clear that one can go back and forth between the language of fibred $$\infty$$-categories and the one of algebraic derivators."

By Bal19 he intends Balzin's Reedy model structures in families. Let me quote the abovementioned Theorem 2.

Thm. 2 in Bal19.

Let $$\mathsf{E} \to \mathsf{R}$$ be a left model Reedy fibration. Then the induced $$\infty$$-functor $$\mathsf{L}\text{Sect}(\mathsf{R}, \mathsf{E}) \to \text{Sect}(\mathsf{R}, \mathsf{LE})$$ is an equivalence of quasicategories.

Even though I sense a connection between derivators, fibred $$\infty$$-categories and this statement, I cannot make it precise. Could someone help me spelling out what Cisinski means precisely?

Let $$\mathcal E \to \mathcal C$$ be a Quillen presheaf (model categorical fibres, Quillen pairs as transition functors) where each $$\mathcal E(c)$$ is stable. This would imply, if we had the cofibrant generation in fibres for example, that for any functor $$F: \mathcal D \to \mathcal C$$, the category of sections $$Sect(\mathcal D,F^*\mathcal E)$$ is a stable model category. You can then introduce a $$2$$-functor $$\mathbb D: (Cat/ \mathcal C)^{op} \to CAT$$ that sends $$F: \mathcal D \to \mathcal C$$ to the homotopy category $$Ho (Sect(\mathcal D,F^*\mathcal E))$$ and study its properties. Something like that could be called a relative (pre)derivator (maybe it already is, forgive my ignorance, then) valued in triangulated categories.
We can also start with the infinity-localisation $$L \mathcal E \to \mathcal C$$ and study its infinity-categorical sections $$Sect(\mathcal D,F^* L \mathcal E)$$ for functors $$F: \mathcal D \to \mathcal C$$. Since a localisation of a stable model category is a stable infinity-category (a statement true even without cofibrant generation due to Proposition 3 mentioned below), $$L \mathcal E \to \mathcal C$$ is a bicartesian fibration with stable fibres and using Section 5 of HTT (co/limits of sections computed fibrewise) we can verify that each $$Sect(\mathcal D,F^* L \mathcal E)$$ is stable. Taking their homotopy categories would again yield another $$2$$-functor $$\mathbb D':(Cat/ \mathcal C)^{op} \to CAT$$ valued in triangulated categories.
Proposition 3 of the Reedy paper means however that $$Ho (Sect(\mathcal D,F^*\mathcal E)) \cong Ho (Sect(\mathcal D,F^* L \mathcal E))$$ where the latter is the homotopy category of an infinity-category and the former is that of a model category. In other words, the relative derivators $$\mathbb D$$ and $$\mathbb D'$$ are equivalent; the equivalence comes from the fairly canonical map $$L Sect(\mathcal D,F^*\mathcal E)) \to Sect(\mathcal D,F^* L \mathcal E)$$ (universality of localisation is used here) so it should be trackable along base changes. Thus you can study your relative derivators any way you want, via the model-categorical presentation or the infinity-categorical one.
I suppose you get an algebraic derivator if you replace $$\mathcal C$$ by $$Sch$$ (no restriction on the size of $$\mathcal C$$ was made above) and consider something like $$QCoh \to Sch$$ for the Quillen presheaf, but you will have to take it from here to see if that fits. Maybe I should add something about such relative derivators to the Reedy paper if life permits.