Dynamic (also known as Benamou-Brenier) formulations of optimal transport (OT) and its extension to the unbalanced setting, frame the task of comparing two given positive measures as a variational problem in which one searches for measure paths that minimize a certain kinetic energy functional. In applications, it is frequently natural to require such paths to satisfy additional conditions. Inspired by this, we introduce a model which incorporates, into the framework of unbalanced OT, a particular family of integral equality and inequality constraints on the admissible paths of the distribution, transport field and/or source term. We examine the well-posedness of the resulting class of convex variational problems and the properties of the induced metrics on the corresponding measure subspaces. We will then consider a discretized version of those problems and derive an adaptation of the parallel proximal scheme from convex optimization to numerically approximate solutions.