Ribbon Concordance on 3-Manifolds
This post is a guess. Ribbon concordance on 3-manifolds can be separated into partial order systems associated with each element of the fundamental group of the 3-manifold. For each element of the fundamental group, a partial order system can be associated. The reason behind the guess is that if the knotty part can shrink to very small and the remaining part is largely isotopic to a representative element of the fundamental group, then it's likely that the structure obtained in 3-sphere case is applicable to each element of the fundamental group. It's a interesting research topic.