The conventional momentum-exchange method (CME) is verified to be accurate for the stationary boundary by Mei et al. [Phys. Rev. E 65, 041203 (2002)], but it might be inaccurate when the boundary is moving in the lattice Boltzmann simulations. A lattice-type-dependent momentum-exchange method (LME) is presented to evaluate the hydrodynamic force on moving boundaries, in which the additional momenta induced by the type-changing lattices are well considered. LME preserves the superior features of CME, such as reliability, simplicity, and parallelism. Without any interpolation and integration, the algorithm is independent of boundary geometries, and therefore, efficient in computation and easy to be implemented in both two and three dimensions. A series of cylinder sedimentations are simulated to illustrate the accuracy and robustness of LME, and the results are in excellent agreement with those by the arbitrary Lagrangian-Eulerian technique (ALE). The lateral migrations of a particle are also investigated in the simulations of a neutrally buoyant cylinder in a Poiseuille flow, and consistent with the Segré-Silberberg effect.