Conditional diffusion models appear capable of compositional generalization, i.e., generating convincing samples for out-of-distribution combinations of conditioners, but the mechanisms underlying this ability remain unclear. To make this concrete, we study length generalization, the ability to generate images with more objects than seen during training. In a controlled CLEVR setting (Johnson et al., 2017), we find that length generalization is achievable in some cases but not others, suggesting that models only sometimes learn the underlying compositional structure. We then investigate locality as a structural mechanism for compositional generalization. Prior works proposed score locality as a mechanism for creativity in unconditional diffusion models (Kamb & Ganguli, 2024; Niedoba et al., 2024), but did not address flexible conditioning or compositional generalization. In this paper, we prove an exact equivalence between a specific compositional structure (“conditional projective composition”) (Bradley et al., 2025) and scores with sparse dependencies on both pixels and conditioners (“local conditional scores”). This theory also extends to feature-space compositionality. We validate our theory empirically: CLEVR models that succeed at length generalization exhibit local conditional scores, while those that fail do not. Furthermore, we show that a causal intervention explicitly enforcing local conditional scores restores length generalization in a previously failing model. Finally, we investigate feature-space compositionality in color-conditioned CLEVR, and find preliminary evidence of compositional structure in SDXL.