In multisite trials, learning about treatment effect variation across sites is critical for understanding where and for whom a program works. Unadjusted comparisons, however, capture "compositional" differences in the distributions of unit-level features as well as "contextual" differences in site-level features, including possible differences in program implementation. Our goal in this article is to adjust site-level estimates for differences in the distribution of observed unit-level features: If we can reweight (or "transport") each site to have a common distribution of observed unit-level covariates, the remaining treatment effect variation captures contextual and unobserved compositional differences across sites. This allows us to make apples-to-apples comparisons across sites, parceling out the amount of cross-site effect variation explained by systematic differences in populations served. In this article, we develop a framework for transporting effects using approximate balancing weights, where the weights are chosen to directly optimize unit-level covariate balance between each site and the common target distribution. We first develop our approach for the general setting of transporting the effect of a single-site trial. We then extend our method to multisite trials, assess its performance via simulation, and use it to analyze a series of multisite trials of adult education and vocational training programs. In our application, we find that distributional differences are potentially masking cross-site variation. Our method is available in the balancer R package. [This is the online version of an article published in "Journal of Educational and Behavioral Statistics."]