Dynamics of the imbalance in occupations on even and odd sites of a lattice serves as one of the key characteristics for the identification of the many-body localization transition. In this work, we investigate the long-time behavior of the imbalance in disordered one- and two-dimensional many-body systems in the regime of diffusive or subdiffusive transport. We show that memory effects originating from a coupling between slow and fast modes lead to a power-law decay of the imbalance, with the

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... xponent determined by the diffusive (or subdiffusive) transport law and the spatial dimensionality. Analytical results are supported by numerical simulations performed on a two-dimensional system in the regime of weak localization.