Transport maps deterministically link a complex target distribution to a simpler reference, enabling explicit density evaluation and fast independent sampling. Triangular transport maps, with their efficient Jacobian structure and strong theoretical guarantees, have been successfully deployed for Bayesian inference, generative modeling, and beyond---but learning them via inverse transport usually demands many costly, high-fidelity samples, which can be untenable when sampling the target distribution is computationally expensive. We overcome this bottleneck by exploiting multifidelity information: a small set of high-fidelity samples and larger pools of cheaper-to-obtain lower-fidelity samples coming from distributions related to the high-fidelity target. We further consider a more challenging setting where the data at different fidelity levels are federated; i.e., they reside on separate clients. Within the developed methods no raw data are shared---only map parameters or gradient updates. We propose two federated inverse‐transport strategies: (i) Hierarchical deep transport: train local triangular maps between successive fidelity levels and compose them into a global coupling. (ii) Non-hierarchical multifidelity transport: fit one global triangular map augmented with multifidelity corrective terms, using federated gradient aggregation. We deploy the developed methods on a variety of benchmark problems that illuminate under what conditions high-fidelity data requirements can be successfully reduced.