The pseudo-effective cone is the closure of the convex cone of effective cycle classes of arbitrary fixed (co)dimension in the numerical group of a projective variety. Of particular interest are the boundary classes that were added upon closing. A recent conjecture by Debarre, Jiang, and Voisin aims to bring geometry to such classes that pushforward to zero by a morphism to another projective variety. I will present progress on this and related results coming from joint work with Brian Lehmann.