Descent theorems for fundamental groups and groupoids? Grothendieck in his 1984 "Esquisse d'un programme" (Section 2) wrote (English translation):
" ..,people still obstinately persist,  when calculating with fundamental groups, in fixing a single base point, instead of cleverly choosing a whole packet of points which is invariant under  the symmetries of the situation, which thus get lost on the way.  In certain situations (such as descent theorems for fundamental groups `a la van  Kampen) it is much more elegant,  even indispensable for understanding  something, to work with fundamental groupoids with respect to a suitable    packet of base points,.."
Question: Does anyone have any reference to relevant work on "descent theorems for fundamental groups"? 
Relevant to this question is this mathoverflow  discussion on several base points.  
 A: 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. 
