Before revision:  These methods imply that linearly adjusting for differences between clusters in average covariate values addresses all concerns with cross-cluster comparisons. Instead, we consider an exponential family structure on the within-cluster distribution of covariates and treatments that implies that a low-dimensional sufficient statistic can summarize the empirical distribution, where this sufficient statistic may include functions of the data beyond average covariate values.
After revision:  These methods imply that linearly adjusting for differences between clusters in average covariate values addresses all concerns with cross-group comparisons. We start by making two observations. First we note that the fixed effect method in effect adjusts only for differences between groups by adjusting for the average of covariate values and average treatment. Second, we note that weighting by the inverse of the propensity score would remove biases for comparisons between treated and control units under the fixed effect set up. We then develop three generalizations of the fixed effect approach based on these two observations. First, we suggest more general, nonlinear, adjustments for the average covariate values.