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Many medical datasets have recently been created for medical image segmentation tasks, and it is natural to question whether we can use them to sequentially train a single model that (1) performs better on all these datasets, and (2) generalizes well and transfers better to the unknown target site domain. Prior works have achieved this goal by jointly training one model on multi-site datasets, which achieve competitive performance on average but such methods rely on the assumption about the availability of all training data, thus limiting its effectiveness in practical deployment. In this paper, we propose a novel multi-site segmentation framework called incremental-transfer learning (ITL), which learns a model from multi-site datasets in an end-to-end sequential fashion. Specifically, "incremental" refers to training sequentially constructed datasets, and "transfer" is achieved by leveraging useful information from the linear combination of embedding features on each dataset. In addition, we introduce our ITL framework, where we train the network including a site-agnostic encoder with pre-trained weights and at most two segmentation decoder heads. We also design a novel site-level incremental loss in order to generalize well on the target domain. Second, we show for the first time that leveraging our ITL training scheme is able to alleviate challenging catastrophic forgetting problems in incremental learning. We conduct experiments using five challenging benchmark datasets to validate the effectiveness of our incremental-transfer learning approach. Our approach makes minimal assumptions on computation resources and domain-specific expertise, and hence constitutes a strong starting point in multi-site medical image segmentation.