Puncture-forgetful maps have been studied in various settings, including Teichmüller spaces, mapping class groups, and spaces of closed curves. However, puncture-forgetful maps are known to be discontinuous for curves and for the Thurston boundary (i.e., projective measured foliations). In this talk, we introduce several approaches to forgetting punctures in measured foliations that yield upper semi-continuous maps. We then discuss applications to Teichmüller geodesics, the dynamics of postcritically finite rational maps, and point-pushing mapping classes. This is joint work with Jeremy Kahn.