When a suspension of platelet-like particles sediment in a closed container, the particles undergo isotropic-nematic phase transition (I-N transition), and there appears a clear interface between the isotropic phase and the nematic phase. Usually the interface moves from bottom to top since the nematic phase appears and grows at the bottom, but it has been observed that in some situations the interface moves from top to bottom. Here, we study the dynamics of the interface by solving the non-equilibrium diffusion equation for the concentration of platelet-like particles, and show that the I-N interface can move upward (rising interface) or downward (falling interface) depending on whether the initial concentration is less than the critical concentration of I-N transition or more than it. We give a simple analysis theory for the motion of the interface in each case, which agrees well with the numerical calculations. We also show that the numerical results are in reasonable agreement with existing experimental measurements.