The paper presents a novel three-step transfer learning framework for enhancing cross-lingual transfer from high- to low-resource languages in the downstream application of Automatic Speech Translation. The approach integrates a semantic knowledge-distillation step into the existing two-step cross-lingual transfer learning framework XLS-R. This extra step aims to encode semantic knowledge in the multilingual speech encoder pre-trained via Self-Supervised Learning using unlabeled speech. Our proposed three-step cross-lingual transfer learning framework addresses the large cross-lingual transfer gap (TRFGap) observed in the XLS-R framework between high-resource and low-resource languages. We validate our proposal through extensive experiments and comparisons on the CoVoST-2 benchmark, showing significant improvements in translation performance, especially for low-resource languages, and a notable reduction in the TRFGap.