The mechanisms by which chemical vapor deposited (CVD) graphene and hexagonal boron nitride (h-BN) films can be released from a growth catalyst, such as widely used copper (Cu) foil, are systematically explored as a basis for an improved lift-off transfer. We show how intercalation processes allow the local Cu oxidation at the interface followed by selective oxide dissolution, which gently releases the 2D material (2DM) film. Interfacial composition change and selective dissolution can thereby be achieved in a single step or split into two individual process steps. We demonstrate that this method is not only highly versatile but also yields graphene and h-BN films of high quality regarding surface contamination, layer coherence, defects, and electronic properties, without requiring additional post-transfer annealing. We highlight how such transfers rely on targeted corrosion at the catalyst interface and discuss this in context of the wider CVD growth and 2DM transfer literature, thereby fostering an improved general understanding of widely used transfer processes, which is essential to numerous other applications.