Randomized controlled trials (RCTs) are the gold standard for evaluating causal effects but are often costly and difficult to scale; consequently, they are frequently augmented with auxiliary external controls in many applications. Prior approaches to borrowing such data typically rely on exchangeability, under which the external controls are readily usable for inference in the trial population. In practice, however, differences in eligibility criteria, standard of care, and data collection procedures may induce distribution shifts between the RCT and the external controls, making exchangeability implausible. In this paper, we propose a general framework for incorporating external controls that explicitly accounts for distribution shifts through an augmented estimating function, allowing information about the shifts to be leveraged for efficiency gains even when exchangeability does not hold. To guard against potential bias and efficiency loss due to misspecification of the distribution shift, we further develop an adaptive shrinkage estimator that preserves consistency while guaranteeing efficiency dominance over the trial-only benchmark. Synthetic experiments and a real data application demonstrate the practical advantages of the proposed approaches.

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