@Adeel Thanks, but neither of these is quite what I need. I need index diagrams that are more complicated than just products (so 5.5.8.11 isn't enough), and 5.3.3.3 is only about limits/colimits in the category of spaces (I need the category of stable infty-categories). However, there is something about my case that's not that far from the category of spaces, and I'm editing the question accordingly.

Sorry! The fact that $f|_{I^n_X}$ is the identity on the first $r$ coordinates doesn't imply that its image is $I^r\times U$. (@user52824, the incorrect arguments I gave are in fact local).

@Alex: I thought any group homology of a finitely presented group is finite-dimensional. Are there examples where $H_2$ isn't? And yes, "infinite" means infinite-dimensional

There are probably no modifications necessary in this case - thanks. And I'm interested in the torsion-free part, so fine with taking coefficients in any characteristic-zero field. Edited question accordingly.

@David So it sounds like you're saying (in the algebraic case), that the 'etale topos contains all the topology of a manifold up to some sort of profinite completion, and higher-category analogues of locally constant (resp. constructible) sheaves are well-approximated by sheaves on this topos on the one hand, and therefore by D-modules on the other hand. Is this correct?