The Riemann existence theorem precisely describes branched covers of Riemann surfaces in terms of the monodromy data of the covers. The Grothendieck school was able to apply these transcendental methods to the context of branched covers of algebraic curves in characteristic $p$, but much remains open in the case that the order of the monodromy group is not prime to $p$. We discuss various aspects of approaching this problem via degeneration techniques, emphasizing the broader philosophy. We include new results obtained by shifting perspective from branched covers to linear series, issues regarding braid group actions and dependence on a choice of generators of the fundamental group, and some general discussion of what one might hope to accomplish via more direct degenerations.