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Beamforming is a powerful tool designed to enhance speech signals from the direction of a target source. Computing the beamforming filter requires estimating spatial covariance matrices (SCMs) of the source and noise signals. Time-frequency masks are often used to compute these SCMs. Most studies of mask-based beamforming have assumed that the sources do not move. However, sources often move in practice, which causes performance degradation. In this paper, we address the problem of mask-based beamforming for moving sources. We first review classical approaches to tracking a moving source, which perform online or blockwise computation of the SCMs. We show that these approaches can be interpreted as computing a sum of instantaneous SCMs weighted by attention weights. These weights indicate which time frames of the signal to consider in the SCM computation. Online or blockwise computation assumes a heuristic and deterministic way of computing these attention weights that, although simple, may not result in optimal performance. We thus introduce a learning-based framework that computes optimal attention weights for beamforming. We achieve this using a neural network implemented with self-attention layers. We show experimentally that our proposed framework can greatly improve beamforming performance in moving source situations while maintaining high performance in non-moving situations, thus enabling the development of mask-based beamformers robust to source movements.