We consider the problem of extrapolating treatment effects across heterogeneous populations (``sites"/``contexts"). We consider an idealized scenario in which the researcher observes cross-sectional data for a large number of units across several ``experimental" sites in which an intervention has already been implemented to a new ``target" site for which a baseline survey of unit-specific, pre-treatment outcomes and relevant attributes is available. Our approach treats the baseline as functional data, and this choice is motivated by the observation that unobserved site-specific confounders manifest themselves not only in average levels of outcomes, but also how these interact with observed unit-specific attributes. We consider the problem of determining the optimal finite-dimensional feature space in which to solve that prediction problem. Our approach is design-based in the sense that the performance of the predictor is evaluated given the specific, finite selection of experimental and target sites. Our approach is nonparametric, and our formal results concern the construction of an optimal basis of predictors as well as convergence rates for the estimated conditional average treatment effect relative to the constrained-optimal population predictor for the target site. We quantify the potential gains from adapting experimental estimates to a target location in an application to conditional cash transfer (CCT) programs using a combined data set from five multi-site randomized controlled trials.