Using a combination of complementary in situ X-ray photoelectron spectroscopy and X-ray diffraction, we study the fundamental mechanisms underlying the chemical vapor deposition (CVD) of hexagonal boron nitride (h-BN) on polycrystalline Cu. The nucleation and growth of h-BN layers is found to occur isothermally, i.e., at constant elevated temperature, on the Cu surface during exposure to borazine. A Cu lattice expansion during borazine exposure and B precipitation from Cu upon cooling highlight that B is incorporated into the Cu bulk, i.e., that growth is not just surface-mediated. On this basis we suggest that B is taken up in the Cu catalyst while N is not (by relative amounts), indicating element-specific feeding mechanisms including the bulk of the catalyst. We further show that oxygen intercalation readily occurs under as-grown h-BN during ambient air exposure, as is common in further processing, and that this negatively affects the stability of h-BN on the catalyst. For extended air exposure Cu oxidation is observed, and upon re-heating in vacuum an oxygen-mediated disintegration of the h-BN film via volatile boron oxides occurs. Importantly, this disintegration is catalyst mediated, i.e., occurs at the catalyst/h-BN interface and depends on the level of oxygen fed to this interface. In turn, however, deliberate feeding of oxygen during h-BN deposition can positively affect control over film morphology. We discuss the implications of these observations in the context of corrosion protection and relate them to challenges in process integration and heterostructure CVD.