Using the combination of experiment and molecular dynamics simulations, we investigate structural transformations in magnetic elastomers with NdFeB flake-like particles, caused by applied moderate magnetic fields. We explain why and how those transformations depend on whether or not the samples are initially cured by a short-time exposure to a strong field. We find that in a cured sample, a moderate magnetic field leads mainly to in-place flake rotations that are fully reversed once the applied field is switched off. In contrast, in an initially non-cured sample the flakes perform both translation and rotations under the influence of a moderate applied field that lead to the formation of chain-like structures that remain such even if the field is switched off.