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We consider the problem of fairly allocating items to a set of individuals, when the items are arriving online. A central solution concept in fair allocation is competitive equilibrium: every individual is endowed with a budget of faux currency, and the resulting competitive equilibrium is used to allocate. For the online fair allocation context, the PACE algorithm of Gao et al. [2021] leverages the dual averaging algorithm to approximate competitive equilibria. The authors show that, when items arrive i.i.d, the algorithm asymptotically achieves the fairness and efficiency guarantees of the offline competitive equilibrium allocation. However, real-world data is typically not stationary. One could instead model the data as adversarial, but this is often too pessimistic in practice. Motivated by this consideration, we study an online fair allocation setting with nonstationary item arrivals. To address this setting, we first develop new online learning results for the dual averaging algorithm under nonstationary input models. We show that the dual averaging iterates converge in mean square to both the underlying optimal solution of the "true" stochastic optimization problem as well as the "hindsight" optimal solution of the finite-sum problem given by the sample path. Our results apply to several nonstationary input models: adversarial corruption, ergodic input, and block-independent (including periodic) input. Here, the bound on the mean square error depends on a nonstationarity measure of the input. We recover the classical bound when the input data is i.i.d. We then show that our dual averaging results imply that the PACE algorithm for online fair allocation simultaneously achieves "best of both worlds" guarantees against any of these input models. Finally, we conduct numerical experiments which show strong empirical performance against nonstationary inputs.