I think I can now reasonably point to the book

Bourbaki, Topologie Algébrique: Chapitres 1 à 4 (Elements de Mathématique) 7 Apr 2016

which does use ideas of descent and the fundamental groupoid both for the van Kampen theorem and for discrete groups acting properly on spaces. They call the fundamental groupoid the "Poincaré groupoid", though I think its topological use goes back only to Reidemeister's 1932 book.

Their Theorem 3 on p. 419 seems to overlap with the work on orbit groupoids of Chapter 11 of Topology and Groupoids, which gives the application to the symmetric square of a space.

Comments welcome!

Aug 21, 2016 I should mention the paper

"Van Kampen theorems for categories of covering morphisms in lextensive categories" R. Brown, G. Janelidze, J. Pure Applied Algebra I 19 (1997) 255-283. (pdf)

This considers the whole fundamental groupoid, not "many base points", but does use descent notions in general situations.

I think I can now reasonably point to the book

Bourbaki, Topologie Algébrique: Chapitres 1 à 4 (Elements de Mathématique) 7 Apr 2016

which does use ideas of descent and the fundamental groupoid both for the van Kampen theorem and for discrete groups acting properly on spaces. They call the fundamental groupoid the "Poincaré groupoid", though I think its topological use goes back only to Reidemeister's 1932 book. However, despite the comment of Grothendieck, they do not use the fundamental groupoid on a set of base points.

Their Theorem 3 on p. 419 seems to overlap with the work on orbit groupoids of Chapter 11 of Topology and Groupoids, which gives the application to the symmetric square of a space.

Comments welcome!

Aug 21, 2016 I should mention the paper

"Van Kampen theorems for categories of covering morphisms in lextensive categories" R. Brown, G. Janelidze, J. Pure Applied Algebra I 19 (1997) 255-283. (pdf)

This considers the whole fundamental groupoid, not "many base points", but does use descent notions in general situations.

I think I can now reasonably point to the book

Bourbaki, Topologie Algébrique: Chapitres 1 à 4 (Elements de Mathématique) 7 Apr 2016

which does use ideas of descent and the fundamental groupoid both for the van Kampen theorem and for discrete groups acting properly on spaces. They call the fundamental groupoid the "Poincaré groupoid", though I think its topological use goes back only to Reidemeister's 1932 book.

Their Theorem 3 on p. 419 seems to overlap with the work on orbit groupoids of Chapter 11 of Topology and Groupoids, which gives the application to the symmetric square of a space.

Comments welcome!

I think I can now reasonably point to the book

Bourbaki, Topologie Algébrique: Chapitres 1 à 4 (Elements de Mathématique) 7 Apr 2016

which does use ideas of descent and the fundamental groupoid both for the van Kampen theorem and for discrete groups acting properly on spaces. They call the fundamental groupoid the "Poincaré groupoid", though I think its topological use goes back only to Reidemeister's 1932 book.

Their Theorem 3 on p. 419 seems to overlap with the work on orbit groupoids of Chapter 11 of Topology and Groupoids, which gives the application to the symmetric square of a space.

Comments welcome!

Aug 21, 2016 I should mention the paper

"Van Kampen theorems for categories of covering morphisms in lextensive categories" R. Brown, G. Janelidze, J. Pure Applied Algebra I 19 (1997) 255-283. (pdf)

This considers the whole fundamental groupoid, not "many base points", but does use descent notions in general situations.