Transport phenomena for different boundary conditions in finite and effectively semi-infinite domains were successfully simulated using the lattice Boltzmann method (LBM). We verified an LBM solute transport algorithm and its boundary conditions by comparing simulation results with analytical solutions for four different one-dimensional solute transport problems. Zero-diffusion analytical breakthrough curves were derived for parallel plate and pipe flow, and LBM simulations with small diffusion coefficients matched these well. Simulations of solute transport in one-dimensional finite and semi-infinite domains were performed at a low column Peclet number (Pe = 1) for first- and third-type inlet boundary conditions; these followed the analytical solutions closely. A series of transport simulations were performed to demonstrate the impact of diffusion and dispersion on the solute front. Taylor dispersion coefficients for the simulated range of Peclet numbers were estimated using moment analysis of the concentration obtained from transport simulation by the LBM. The simulation results showed good agreement with theoretical predictions and thus verified the robustness of LBM-based transport models.