We introduce a new model for conditional and continuous data morphing called TRansport Adversarial Network for Smooth InTerpolation (TRANSIT). We apply it to create a background data template for weakly-supervised searches at the LHC. The method smoothly transforms sideband events to match signal region mass distributions. We demonstrate the performance of TRANSIT using the LHC Olympics R&D dataset. The model captures non-linear mass correlations of features and produces a template that offers a competitive anomaly sensitivity compared to state-of-the-art transport-based template generators. Moreover, the computational training time required for TRANSIT is an order of magnitude lower than that of competing deep learning methods. This makes it ideal for analyses that iterate over many signal regions and signal models. Unlike generative models, which must learn a full probability density distribution, i.e., the correlations between all the variables, the proposed transport model only has to learn a smooth conditional shift of the distribution. This allows for a simpler, more efficient residual architecture, enabling mass uncorrelated features to pass the network unchanged while the mass correlated features are adjusted accordingly. Furthermore, we show that the latent space of the model provides a set of mass decorrelated features useful for anomaly detection without background sculpting.