Connectedness of families of sphere covers of a given type

There are now many applications of the following basic problem: Do all covers of the sphere by a compact Riemann surface of a “given type” compose one connected family? Or failing that, do they fall into easily discernible components? The meaning of “given type” usually uses the idea of a Nielsen class — a concept for covers that generalizes the genus of a compact Riemann surface. The answer has often been yes, and that answer has figured in many problems from the connectedness of the moduli space of curves of genus g (geometry) to Davenport’s problem (arithmetic) and the genus 0 problem (group theory). This survey arose in response to the following special case asked by Brian Osserman. Do all genus zero covers of the sphere with r specific pure-cycles as branch cycles form one connected family?