While accuracy is an evident criterion for ultrasound image segmentation, output consistency across different tests is equally crucial for tracking changes in regions of interest in applications such as monitoring the patients' response to treatment, measuring the progression or regression of the disease, reaching a diagnosis, or treatment planning. Convolutional neural networks (CNNs) have attracted rapidly growing interest in automatic ultrasound image segmentation recently. However, CNNs are not shift-equivariant, meaning that if the input translates, e.g., in the lateral direction by one pixel, the output segmentation may drastically change. To the best of our knowledge, this problem has not been studied in ultrasound image segmentation or even more broadly in ultrasound images. Herein, we investigate and quantify the shift-variance problem of CNNs in this application and further evaluate the performance of a recently published technique, called BlurPooling, for addressing the problem. In addition, we propose the Pyramidal BlurPooling method that outperforms BlurPooling in both output consistency and segmentation accuracy. Finally, we demonstrate that data augmentation is not a replacement for the proposed method. Source code is available at https://git.io/pbpunet and http://code.sonography.ai.