FST and kinship are key parameters often estimated in modern population genetics studies. Kinship matrices have also become a fundamental quantity used in genome-wide association studies and heritability estimation. The most frequently used estimators of FST and kinship are method of moments estimators whose accuracies depend strongly on the existence of simple underlying forms of structure, such as the island model of non-overlapping, independently evolving subpopulations. However, modern data sets have revealed that these simple models of structure do not likely hold in many populations, including humans. In this work, we provide new results on the behavior of these estimators in the presence of arbitrarily complex population structures. After establishing a framework for assessing bias and consistency of genome-wide estimators, we calculate the accuracy of FST and kinship estimators under arbitrary population structures, characterizing biases and estimation challenges unobserved under their originally assumed models of structure. We illustrate our results using simulated genotypes from an admixture model, constructing a one-dimensional geographic scenario that departs nontrivially from the island model. Using 1000 Genomes Project data, we verify that population-level pairwise FST estimates underestimate differentiation measured by an individual-level pairwise FST estimator introduced here. We show that the calculated biases are due to unknown quantities that cannot be estimated under the established frameworks, highlighting the need for innovative estimation approaches in complex populations. We provide initial results that point towards a future estimation framework for generalized FST and kinship.