If an ultrasound imaging system provides a presentation of a moving object which is sensitive to small spatial shifts, the system is said to be locally spatially shift-variant. This can happen, for instance, if the axial or lateral sampling is insufficient. The Capon beamformer has been shown to provide increased lateral resolution in ultrasound images. Increased lateral resolution should demand denser lateral sampling. However, in previous literature on Capon beamforming for medical ultrasound imaging, only single-frame scenarios have been simulated. Temporal behavior and effects caused by the increased resolution and lack of oversampling have therefore been neglected. In this paper, we analyze the local lateral shift-invariance of the Capon beamformer when imaging moving objects. We show that insufficient lateral sampling makes an imaging system based on the Capon beamformer laterally shift-variant. Different methods for oversampling on transmit and receive are then discussed and investigated to improve on the Capon beamformer. It is shown that lateral shift-invariance can be improved by oversampling based on phase rotation on receive without affecting the acquisition frame rate and with a minor change in processing complexity.