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While many deep learning methods have seen significant success in tackling the problem of domain adaptation and few-shot learning separately, far fewer methods are able to jointly tackle both problems in Cross-Domain Few-Shot Learning (CD-FSL). This problem is exacerbated under sharp domain shifts that typify common computer vision applications. In this paper, we present a novel, flexible and effective method to address the CD-FSL problem. Our method, called Score-based Meta Transfer-Learning (SB-MTL), combines transfer-learning and meta-learning by using a MAML-optimized feature encoder and a score-based Graph Neural Network. First, we have a feature encoder with specific layers designed to be fine-tuned. To do so, we apply a first-order MAML algorithm to find good initializations. Second, instead of directly taking the classification scores after fine-tuning, we interpret the scores as coordinates by mapping the pre-softmax classification scores onto a metric space. Subsequently, we apply a Graph Neural Network to propagate label information from the support set to the query set in our score-based metric space. We test our model on the Broader Study of Cross-Domain Few-Shot Learning (BSCD-FSL) benchmark, which includes a range of target domains with highly varying dissimilarity to the miniImagenet source domain. We observe significant improvements in accuracy across 5, 20 and 50 shot, and on the four target domains. In terms of average accuracy, our model outperforms previous transfer-learning methods by 5.93% and previous meta-learning methods by 14.28%.