Coded caching is a valuable technique for reducing transmission rates in cache-enabled broadcast networks by leveraging coding and multicast opportunities. The standard Maddah-Ali-Niesen scheme has an order-optimal rate but suffers from subpacketization exponential in the number of users for certain memory regimes. High subpacketization, which involves dividing files into numerous subfiles, is unsuitable for typical file sizes and introduces high computational overhead. In contrast, coded caching schemes designed using placement delivery arrays (PDAs) can have linear subpacketization with a penalty in rate. We present lifting constructions for PDAs, where smaller PDAs are combined to create larger PDAs. Blackburn compatibility, a new notion for PDAs, facilitates these lifting constructions. We present diverse families of Blackburn-compatible PDAs, including algebraic and randomized constructions. Numerical evaluations demonstrate the good performance of our schemes for a wide range of parameters, simultaneously achieving low subpacketization. Furthermore, we demonstrate the versatility of lifting constructions by casting some existing coded caching schemes within this framework.