Given a holomorphic family of Riemann surfaces is it possible to associate a holomorphically varying finite collection of points to each Riemann surface in the family? Hubbard showed that when the family is the entire moduli space of genus g Riemann surfaces this is possible only when g = 2 and the marked points are fixed points of the hyperelliptic involution. We will pose and resolve analogous questions for strata of translation surfaces with marked points. We will draw connections between GL(2,R)-invariant families of marked points on affine invariant submanifolds and holomorphically varying collections of points on closed totally geodesic families of Riemann surfaces. Finally we will discuss applications to billiard problems, specifically the finite blocking and illumination problems.