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Distributed algorithms are derived for estimation and smoothing of nonstationary dynamical processes based on correlated observations collected by ad hoc wireless sensor networks (WSNs). Specifically, distributed Kalman filtering (KF) and smoothing schemes are constructed for any-time minimum mean-square error (MMSE) optimal consensus-based state estimation using WSNs. The novel distributed filtering/smoothing approach is flexible to trade-off estimation delay for MSE reduction, while it exhibits robustness in the presence of communication noise. Numerical examples demonstrate the merits of the proposed approach with respect to existing alternatives.