Maybe my answer will not fit exactly your question. What I like very luch is D. Sullivan's use of Artin-Mazur's theory in his proof of Adams'conjecture. What D. Sullivan does is the computation of the étale homotopy type of the classifying space $BU_n$ of the complex unitary group and he does this computation by considering this classifying space as a direct limit of complex Grassmannians:

$$G_{n,k}\cong GL(n+k,\mathbb{C})/(GL(n,\mathbb{C})\times GL(k,\mathbb{C}))$$

Then he analysises the étale homotopy type of $BU_n$ by looking at its associated arithmetic square. What is important in Sullivan's proof of the Adams'conjecture is the understanding of the action of the galois group on the étale homotopy type of $BU_n$ which has a deep impact on Adams operations in $K$-theory. In his MIT notes "Geometric Topology Localization, Periodicity, and Galois Symmetry" also states a conjecture, now a theorem "the Sullivan's conjecture", that has some important implications on the study of the étale homotopy type of real algebraic varieties. Of course all this material can be found in section 5 "Algebraic geometry (étale homotopy type)" of the notes cited above with many examples.

Maybe my answer will not fit exactly your question. What I like very much is D. Sullivan's use of Artin-Mazur's theory in his proof of Adams'conjecture. What D. Sullivan does is the computation of the étale homotopy type of the classifying space $BU_n$ of the complex unitary group and he does this computation by considering this classifying space as a direct limit of complex Grassmannians:

$$G_{n,k}\cong GL(n+k,\mathbb{C})/(GL(n,\mathbb{C})\times GL(k,\mathbb{C}))$$

Then he analysises the étale homotopy type of $BU_n$ by looking at its associated arithmetic square. What is important in Sullivan's proof of the Adams'conjecture is the understanding of the action of the absolute galois group on the étale homotopy type of $BU_n$ which has a deep impact on Adams operations in $K$-theory. In his MIT notes "Geometric Topology Localization, Periodicity, and Galois Symmetry" he also states a conjecture, now a theorem: "the Sullivan's conjecture", that has some important implications on the study of the étale homotopy type of real algebraic varieties. Of course all this material can be found in section 5 "Algebraic geometry (étale homotopy type)" of the notes cited above with many examples.