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Terrestrial planets are thought to experience episode(s) of large-scale melting early in their history. Fractionation during magma-ocean freezing leads to unstable stratification within the related cumulate layers due to progressive iron enrichment upward, but the effects of incremental cumulate overturns during MO crystallization remain to be explored. Here, we use geodynamic models with a moving-boundary approach to study convection and mixing within the growing cumulate layer, and thereafter within the fully crystallized mantle. For fractional crystallization, cumulates are efficiently stirred due to subsequent incremental overturns, except for strongly iron-enriched late-stage cumulates, which persist as a stably stratified layer at the base of the mantle for billions of years. Less extreme crystallization scenarios can lead to somewhat more subtle stratification. In any case, the long-term preservation of at least a thin layer of extremely enriched cumulates with Fe#>0.4, as predicted by all our models, is inconsistent with seismic constraints. Based on scaling relationships, however, we infer that final-stage Fe-rich magma-ocean cumulates originally formed near the surface should have overturned as small diapirs, and hence undergone melting and reaction with the host rock during sinking. The resulting moderately iron-enriched metasomatized/hybrid rock assemblages should have accumulated at the base of the mantle, potentially fed an intermittent basal magma ocean, and be preserved through the present-day. Such moderately iron-enriched rock assemblages can reconcile the physical properties of the large low shear-wave velocity provinces in the presentday lower mantle. Thus, we reveal Hadean melting and rock-reaction processes by integrating magmaocean crystallization models with the seismic-tomography snapshot.