We report results of an investigation of vertical transport in stacked InAs/InGaAs/InP self-assembled quantum dot multi-layers and show evidence of tunneling between quantum dot states in adjacent layers. In the presence of magnetic fields up to 12 T applied parallel to the current, tunneling through Zeeman-split quasi-zero dimensional states is observed. The difference in the g factor of two quantum dots in adjacent layers, which is due to the difference in confinement, is estimated from the data. The experimental value obtained for the difference in g factor is in good agreement with that obtained from a calculation of the quantum dots’ energy levels in the presence of the magnetic field.