An extension of the projection operators method is presented by considering explicit time-dependent projection operators. The usefulness of the present formalism is demonstrated by an investigation of nonadiabatic corrections to the evolution of a many-body system under a slow motion. A theoretical and experimental study of the saturation of nuclear spins dipolar order induced by a slow sample rotation is presented. Theoretically, the master equation of the dipolar order beyond the limit of an adiabatic evolution is established. It is shown how the time dependence of the projection operators is related to saturation of the dipolar order. A formal expression of the saturation rate is derived and its dependence upon the angle between rotation axis and external magnetic field is derived. Comparison with experimental data obtained on polycrystalline adamantane validates our theoretical approach.