We study distributional transportability of treatment effects in a ``cross-site, one-armed target" design, where both treated and control units are observed in a source site, but only control units are observed in a target site. Our object of interest is not estimating the average treatment effect, but recovering the full treated distribution in the target site using transfer knowledge from the source site, while allowing cross-site heterogeneity in measurement systems, observed features, outcome reporting, population composition, and latent contextual factors. We model cross-site heterogeneity through a transformation between the sites that transports the joint feature--outcome distributions. This transformation is learned from comparing the observed control samples in the source and target sites, using an optimal transport criterion. The learned transformation is then applied to the source treated sample to construct a synthetic sample from the target treated distribution. We establish convergence of the resulting synthetic empirical distribution to the target treated distribution. Simulation studies across multiple data-generating scenarios and a real-world application to patient-derived xenograft (PDX) data demonstrate that our framework recovers the full distributional properties of the target treated population.

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