Medical imaging models frequently fail when deployed across hospitals, scanners, populations, or imaging protocols due to domain shift, limiting their clinical reliability. While transfer learning and domain adaptation address such shifts statistically, they often rely on spurious correlations that break under changing conditions. On the other hand, causal inference provides a principled way to identify invariant mechanisms that remain stable across environments. This survey introduces and systematises Causal Transfer Learning (CTL) for medical image analysis. This paradigm integrates causal reasoning with cross-domain representation learning to enable robust and generalisable clinical AI. We frame domain shift as a causal problem and analyse how structural causal models, invariant risk minimisation, and counterfactual reasoning can be embedded within transfer learning pipelines. We studied spanning classification, segmentation, reconstruction, anomaly detection, and multimodal imaging, and organised them by task, shift type, and causal assumption. A unified taxonomy is proposed that connects causal frameworks and transfer mechanisms. We further summarise datasets, benchmarks, and empirical gains, highlighting when and why causal transfer outperforms correlation-based domain adaptation. Finally, we discuss how CTL supports fairness, robustness, and trustworthy deployment in multi-institutional and federated settings, and outline open challenges and research directions for clinically reliable medical imaging AI.