In a previous paper, we reported simulations of the evolution of the magnetic field in neutron star (NS) cores through ambipolar diffusion, taking the neutrons as a motionless uniform background. However, in real NSs, neutrons are free to move, and a strong composition gradient leads to stable stratification (stability against convective motions) both of which might impact on the time-scales of evolution. Here, we address these issues by providing the first long-term two-fluid simulations of the evolution of an axially symmetric magnetic field in a neutron star core composed of neutrons, protons, and electrons with density and composition gradients. Again, we find that the magnetic field evolves towards barotropic 'Grad-Shafranov equillibria', in which the magnetic force is balanced by the degeneracy pressure gradient and gravitational force of the charged particles. However, the evolution is found to be faster than in the case of motionless neutrons, as the movement of charged particles (which are coupled to the magnetic field, but are also limited by the collisional drag forces exerted by neutrons) is less constrained, since neutrons are now allowed to move. The possible impact of non-axisymmetric instabilities on these equilibria, as well as beta decays, proton superconductivity, and neutron superfluidity, are left for future work.