Regularization and data augmentation are crucial to training end-to-end automatic speech recognition systems. Dropout is a popular regularization technique, which operates on each neuron independently by multiplying it with a Bernoulli random variable. We propose a generalization of dropout, called ``convolutional dropout'', where each neuron's activation is replaced with a randomly-weighted linear combination of neuron values in its neighborhood. We believe that this formulation combines the regularizing effect of dropout with the smoothing effects of the convolution operation. In addition to convolutional dropout, this paper also proposes using random wordpiece segmentations as a data augmentation scheme during training, inspired by results in neural machine translation. We adopt both these methods during the training of transformer-transducer speech recognition models, and show consistent improvements over strong baselines across different languages.