Convolutional neural networks (CNNs) offer a promising means to achieve fast deformable image registration with accuracy comparable to conventional, physics-based methods. A persistent question with CNN methods, however, is whether they will be able to generalize to data outside of the training set. We investigated this question of mismatch between train and test data with respect to first- and second-order image statistics (e.g., spatial resolution, image noise, and power spectrum). A UNet-based architecture was built and trained on simulated CT images for various conditions of image noise (dose), spatial resolution, and deformation magnitude. Target registration error was measured as a function of the difference in statistical properties between the test and training data. Generally, registration error is minimized when the training data exactly match the statistics of the test data; however, networks trained with data exhibiting a diversity in statistical characteristics generalized well across the range of statistical conditions considered. Furthermore, networks trained on simulated image content with first- and second-order statistics selected to match that of real anatomical data were shown to provide reasonable registration performance on real anatomical content, offering potential new means for data augmentation. Characterizing the behavior of a CNN in the presence of statistical mismatch is an important step in understanding how these networks behave when deployed on new, unobserved data. Such characterization can inform decisions on whether retraining is necessary and can guide the data collection and/or augmentation process for training.