The problem of integrating multifidelity data has been extensively studied, as an integrated analysis can produce better results than analyzing each type separately. A popular approach is to use a linear autoregressive model with location and scale adjustment parameters. These parameters are usually modeled using stationary Gaussian processes. However, the stationarity assumption may not be appropriate in practice. To introduce nonstationarity for more flexibility, we propose a new integration model that is based on deep Gaussian processes, which can capture nonstationarity by successive warping of latent variables through multiple layers of Gaussian processes. For inference of the proposed model, we derive a doubly stochastic variational inference algorithm. We validate the proposed model using simulated and real-data examples.