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.