I like the following version of SvKT. If $\Pi_1$ is the functor of fundamental groupoid and $(X_i)_{i\in I}$ is a diagram of spaces then $$\Pi_1({\sf hocolim}\: X_i)\simeq {\sf hocolim}\: \Pi_1(X_i).$$

Question: Is there a similar statement for higher homotopy? For example, if we replace $\Pi_1$ by some version of the infinity-groupoid $\Pi_\infty$. But it should be tricky because the homotopy pushout of the diagram $* \leftarrow S^1 \to *$ is $S^2$ whose homotopy groups are complicated.

I know about Brown's version of this theorem but filtered spaces is not a convenient setting for me.