After reviewing the recent heated debate about cosmological embedding and cosmological-driven mass grow of local astrophysical compact objects, I will present a rigorous treatment of the issue in a General Relativity (GR) framework. I will show that while the mass of usual singular black holes (BH) cannot couple to cosmological dynamics, the mass of nonsingular black holes and other compact regular horizonless object (stars) is changed by cosmological expansion. I will find a leading universal linear dependence of the BH mass in terms of the scale factor and subleading power-law terms for horizonless compact objects. I also discuss the important role played in the derivation by the quasi-local Misner-Sharp mass in contrast with the non-local ADM mass. I also briefly comment on the present status of astrophysical observations.