[1]:http://groupoids.org.uk/topgpds.html 
[2]:http://groupoids.org.uk/pdffiles/brown-janelidze-lex.pdf

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][1], 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][2])

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