Many-body localization is a fascinating theoretical concept describing the intricate interplay of quantum interference, i.e., localization, with many-body interaction-induced dephasing. Numerous computational tests and also several experiments have been put forward to support the basic concept. Typically, averages of time-dependent global observables have been considered, such as the charge imbalance. We here investigate within the disordered spinless Hubbard (t -V) model how dephasing manifests in time-dependent variances of observables. We find that after quenching a Ned state the local charge density exhibits strong temporal fluctuations with a damping that is sensitive to disorder W: variances decay in a power-law manner t(-zeta), with an exponent zeta(W) strongly varying with W. A heuristic argument suggests the form zeta approximate to alpha(W)xi(sp), where xi(sp)(W) denotes the noninteracting localization length and alpha(W ) characterizes the multifractal structure of the dynamically active volume fraction of the many-body Hilbert space. In order to elucidate correlations underlying the damping mechanism, exact computations are compared with results from the time-dependent Hartree-Fock approximation. Implications for experimentally relevant observables, such as the imbalance, will be discussed.