Regional climate models (RCMs) are essential tools for simulating and studying regional climate variability and change. However, their high computational cost limits the production of comprehensive ensembles of regional climate projections covering multiple scenarios and driving Global Climate Models (GCMs) across regions. RCM emulators based on deep learning models have recently been introduced as a cost-effective and promising alternative that requires only short RCM simulations to train the models. Therefore, evaluating their transferability to different periods, scenarios, and GCMs becomes a pivotal and complex task in which the inherent biases of both GCMs and RCMs play a significant role. Here we focus on this problem by considering the two different emulation approaches proposed in the literature (PP and MOS, following the terminology introduced in this paper). In addition to standard evaluation techniques, we expand the analysis with methods from the field of eXplainable Artificial Intelligence (XAI), to assess the physical consistency of the empirical links learnt by the models. We find that both approaches are able to emulate certain climatological properties of RCMs for different periods and scenarios (soft transferability), but the consistency of the emulation functions differ between approaches. Whereas PP learns robust and physically meaningful patterns, MOS results are GCM-dependent and lack physical consistency in some cases. Both approaches face problems when transferring the emulation function to other GCMs, due to the existence of GCM-dependent biases (hard transferability). This limits their applicability to build ensembles of regional climate projections. We conclude by giving some prospects for future applications.