  This paper has two main goals. First, we prove nonabelian refinements of
basechange theorems in \'etale cohomology (i.e., prove analogues of the
classical statements for sheaves of spaces). Second, we apply these theorems to
prove a number of results about the \'etale homotopy type. Specifically, we
prove nonabelian refinements of the smooth basechange theorem, Huber-Gabber
affine analogue of the proper basechange theorem, and Fujiwara-Gabber rigidity
theorem. Our methods also recover Chough's nonabelian refinement of the proper
basechange theorem. Transporting an argument of Bhatt-Mathew to the nonabelian
setting, we apply nonabelian proper basechange to show that the profinite
\'etale homotopy type satisfies arc-descent. Using nonabelian smooth and proper
basechange and descent, we give rather soft proofs of a number of K\"unneth
formulas for the \'etale homotopy type.
