Classical pullbacks compose, as is easily checked with the universal property. More precisely, if $\require{AMScd} \begin{CD} A @>>> B\\ @VVV @VVV\\ C @>>> D \end{CD}$ and $\require{AMScd} \begin{CD} B @>>> E\\ @VVV @VVV\\ D @>>> F \end{CD}$ are pullback diagrams, then so is $\require{AMScd} \begin{CD} A @>>> E\\ @VVV @VVV\\ C @>>> F \end{CD}$.
I was wondering whether this was true for homotopy pullbacks, and if so, with what level of generality.
For instance if you take the usual model for homotopy pullbacks in $\mathbf{Top}$, you can get a very concrete homotopy equivalence between the two homotopy pullbacks.
Also if you have a model structure in which, to compute homotopy pullbacks it suffices to replace one map by a fibration and then take the usual pullback, then you can show by checking universal properties that this still works (I was told that perhaps something like "if you have a Reedy model structure on $C^I$ then it works" would work -but I don't know what that means yet)
So the most general setting I can think of, to make sense of the question is : we have a homotopical category $(C,W)$ (that is, a category $C$ with a wide subcategory of weak equivalences, satisfying the 2-out-of-3 or 2-out-of-6 property) and our diagram category $I=\require{AMScd} \begin{CD} &&\bullet\\ & @VVV\\ \bullet @>>> \bullet \end{CD}$ and then $C^I$ is also a homotopical category with pointwise weak equivalences; and we assume $\lim : C^I\to C$ has a right derived functor $\mathbb{R}\lim : C^I\to C$ (with Riehl's terminology in Categorical homotopy theory), then we call $\mathbb{R}\lim (\require{AMScd} \begin{CD} && B\\ & @VVV\\ C @>>> D \end{CD})$ a homotopy pullback of $B,C$ over $D$.
It seems that with this level of generality, I can't get a map from the homotopy pullback to its components (or at least I don't see how) : I get a map in $\mathrm{Ho}(C)$ by the universal property of the Kan extension and by seeing the components as homotopical functors $C^I\to C$, but this map is a priori only a zigzag of maps in $C$; so first of all for the question to make sense we have to figure out a setting in which we do get honest maps from the homotopy pullback to its components, that is, maps in $C$.
I don't really see how to get that so :
What are some natural conditions we can impose on the situation to get natural maps in $C$ from the homotopy pullback to the components of the diagram ? Natural lifts of the natural transformation $\delta\mathbb{R}\lim \implies \delta\pi$ where $\delta : C\to \mathrm{Ho}(C)$ is the localisation functor, and $\pi$ is any of the "component of the diagram" functors ?
Once we have these conditions we can phrase our question :
do homotopy pullbacks compose ? More precisely, if we have two homotopy pullback diagrams as in the very beginning of this question, with the maps $B\to E, D$ and $A\to B,C$ the natural maps induced by the conditions; when is there an isomorphism in $\mathrm{Ho}(C)$ from $A$ to the homotopy pullback of $E,C$ over $F$ making the obvious diagram commute ? When is this isomorphism (or its inverse) a map in $C$ ?
EDIT : See Pece's answer for another formulation of the question: I will also accept answers that answer that other formulation.