We consider the problem of fairly allocating sequentially arriving items to a set of individuals. For this problem, the recently-introduced PACE algorithm leverages the dual averaging algorithm to approximate competitive equilibria and thus generate online fair allocations. PACE is simple, distributed, and parameter-free, making it appealing for practical use in large-scale systems. However, current performance guarantees for PACE require i.i.d. item arrivals. Since real-world data is rarely i.i.d., or even stationary, we study the performance of PACE on nonstationary data. We start by developing new convergence results for the general dual averaging algorithm under three nonstationary input models: adversarially-corrupted stochastic input, ergodic input, and block-independent (including periodic) input. Our results show convergence of dual averaging up to errors caused by nonstationarity of the data, and recover the classical bounds when the input data is i.i.d. Using these results, we show that the PACE algorithm for online fair allocation simultaneously achieves ``best of many worlds'' guarantees against any of these nonstationary input models as well as against i.i.d. input. Finally, numerical experiments show strong empirical performance of PACE against nonstationary inputs.