The problem of integrating multifidelity data has been studied extensively, due to integrated analyses being able to provide better results than separately analyzing various data types. One popular approach is to use linear autoregressive models with location- and scale-adjustment parameters. Such parameters are typically modeled using stationary Gaussian processes. However, the stationarity assumption may not be appropriate in real-world applications. To introduce nonstationarity for enhanced flexibility, we propose a novel integration model based on deep Gaussian processes that can capture nonstationarity via successive warping of latent variables through multiple layers of Gaussian processes. For inference of the proposed model, we use a doubly stochastic variational inference algorithm. We validate the proposed model using simulated and real-data examples.