Inference for locally stationary time series is challenging because the associated hypotheses form an uncountable collection over a continuous time interval, making pointwise false discovery rate (FDR) control inadequate for simultaneous statistical guarantees. We introduce a novel asymptotically uniform false discovery rate (AuFDR), defined as the expectation of the $L_r$-norm of the false discovery proportion (FDP) process where $r$ is allowed to diverge, to quantify and control false discoveries uniformly over time. To operationalize AuFDR control, we develop an inferential framework for time-varying correlations in high-dimensional nonstationary time series that allows for non-Gaussianity, nonlinearity and possible jumps in mean functions. The proposed approach combines robust difference-based estimators with a multiplier-bootstrap procedure to construct uniformly valid time-varying $P$-values. Based on these $P$-values, we propose a time-varying Benjamini--Yekutieli procedure for controlling the AuFDR under arbitrary dependence and establish its asymptotic validity. Extensive simulations demonstrate the finite-sample performance of the proposed method in controlling the AuFDR. Applications to EEG data and financial time-series data illustrate its practical utility.

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