This talk focuses on the question of how identifiability of a mathematical model, that is, whether parameters can be recovered from data, is related to identifiability of its submodels. We look specifically at linear compartmental models -- which arise in many applications including epidemiology and pharmacokinetics -- and investigate whether identifiability is preserved after adding or removing parts of the model. In particular, we examine whether identifiability is preserved when an input, output, edge, or leak is added or deleted. Our results harness standard differential algebraic techniques, so that the question of whether a model is (generically, locally) identifiable becomes equivalent to asking whether the Jacobian matrix of a certain coefficient map, arising from input-output equations, is generically full rank. Along the way, we discover a new combinatorial formula for these input-output equations, and also pose several conjectures. Additionally, we highlight the contributions of four undergraduate and three graduate student co-authors.