We study when iterated vector fields (vector fields composed with themselves) are conservative. We give explicit examples of vector fields for which this self-composition preserves conservatism. Notably, this includes gradient vector fields of loss functions associated to some generalized linear models (including non-convex functions). As we show, characterizing the set of smooth vector fields satisfying this condition yields non-trivial geometric questions. In the context of federated learning, we show that when clients have loss functions whose gradient satisfies this condition, federated averaging is equivalent to gradient descent on a surrogate loss function. We leverage this to derive novel convergence results for federated learning. By contrast, we demonstrate that when the client losses violate this property, federated averaging can yield behavior which is fundamentally distinct from centralized optimization. Finally, we discuss theoretical and practical questions our analytical framework raises for federated learning.