Minimal surfaces in the round 3-sphere enjoy a number of attractive and influential rigidity properties, particularly in low genus. Progress has been made in extending these results to certain analogous settings, such as free boundary minimal surfaces in the Euclidean ball. We will propose perspectives by which these other settings may be studied within a continuous family starting with the round 3-sphere. In this talk, we’ll discuss (half-space) intersection properties, connections with capillary minimal surfaces and other free boundary problems. This is based on joint work (including some in progress) with Keaton Naff.