I will answer my own question, maybe in hope that it is helpful to someone.

Given a functor $X:D \rightarrow Spc$ of $\infty$-categories, we can take the unstraigthening of $X$ (the appropriate generalization of the category of elements of X), as a left fibration $Un(X) \rightarrow D$. It is a standard fact that the colimit over X can be computed as the realization $|Un(X)| = Un(X)[Un(X)^{-1}]$, see e.g. Lurie HTT, Cor 3.3.4.6.

It is also clear that $Un( X^I ) \simeq Un( X )^I$. Hence we have commutativity with products iff

$$(\prod Un( X ) )[ \prod Un( X )^{-1} ]  \rightarrow \prod Un( X )[ Un( X )^{-1} ]$$

is an equivalence (of $\infty$-categories!). This is a classical problem in homotopy theory. The major obstruction is that (higher dimensional) hammock-type diagrams must be bounded in size, i.e. equivalent modulo higher dimensional hammocks to hammocks of a given size constraint. This is for example the case if $Un( X )$ satisfies a functorial 3-arrow calculus.

Alternatively, one can give concrete descriptions of the homotopy groups of a realization of an $\infty$-category via infinite simplicial or cubical subdivisions. The challenge then remains to find control over the size of representing elements.