The rising of the Cyber-Physical System (CPS) and the Industry 4.0 paradigms demands the design and the implementation of Digital Twin Frameworks (DTFs) that may support the quick build of reliable Digital Twins (DTs) for experimental and testing purposes. Most of the current DTF proposals allow generating DTs at a good pace but affect generality, scalability, portability, and completeness. As a consequence, current DTF are mostly domain-specific and hardly span several application domains (e.g., from simple IoT deployments to the modeling of complex critical infrastructures). Furthermore, the generated DTs often requires a high amount of computational resource to run. In this paper, we present LiDiTE, a novel DTF that overcomes the previous limitations by, on the one hand, supporting the building of general-purpose DTs at a fine-grained level, but, on the other hand, with a reduced resource footprint w.r.t. the current state of the art. We show the characteristics of the LiDiTE by building the DT of a complex and real critical infrastructure (i.e., the Smart Poligeneration Microgrid of the Savona Campus) and evaluating its resource consumption. The source code of LiDiTE, as well as the experimental dataset, is publicly available.