Electrohydrodynamic settling of droplets is governed by the combined influence of electric field and gravitational field in many practical applications and innovative researches. To simulate the deformation and motion of the droplet under the interaction of gravity and electric stress, a numerically multiphase model based on the lattice Boltzmann method (LBM) is established. A perfectly dielectric (PD) drop suspended in another immiscible medium with perfectly dielectric fluids and a leaky dielectric (LD) drop immersed in a leaky dielectric medium are considered to study the dynamic behavior of an electrified droplet driven by gravity. The predictions of the LBM model demonstrated in this paper are in good agreement with classic analytical solutions and conventional numerical results reported in previous literature, particularly for drops with small deformation. Moreover, the electrohydrodynamics of settling droplets subjected to a uniform electric field which is perpendicular to a weak gravitational field is studied computationally for fluids with various electrical parameters. Numerical simulations for droplets with different permittivity ratios and conductivity ratios are conducted over a range of electric capillary numbers. It is found that, unlike PD settling droplets which always transform into a prolate shape due to the presence of the electric field, whether the LD droplets under weak inertia deform into a prolate or an oblate shape depends on the electrical properties of the dispersed medium and surroundings. This phenomenon is in a similar manner as in the previous work for EHD droplets owing to the imposition of weak inertia. However, with the shape feature unaltered, the electric stress acted at the fluid-fluid surface has a significant influence on the drop deformation and the distribution of charges and consequently results in noticeable changes in transient settling speed.