The multi-physical transport phenomena through the different size geometries are studied by developing a general local grid refinement approach for lattice Boltzmann methods. Revisiting the method of local fine patches on the coarse grid, through the Chapman-Enskog expansion, the multi-physicochemical source terms such as ion electro-migration, heat source, electric body force, and free net electric charge density can be rigorously incorporated to the rescaling relations of the distribution functions, which interchange between fine and coarse grids. We propose two general local refinement approaches for lattice Boltzmann for momentum and advection-diffusion equations with source terms. To evaluate our algorithm, (i) a body-force driven Poiseuille flow in a channel; (ii) an electro-osmotic flow in which the coupled Poisson-Nernst-Planck with Navier-Stokes equations for overlapped and non-overlapped electric double layers; (iii) a symmetric and asymmetric 1D and 2D heat conduction with heat generation in a flat plate; and (iv) an electric potential distribution near a charged surface, are modeled numerically. Good agreements with the available analytical solutions demonstrated the robustness of the proposed algorithm for diffusion or advection-diffusion equations, which may be coupled or decoupled. The present model may broaden the applications of local grid refinement for modeling complex transport phenomena, such as multi-physicochemical transport phenomena in different size geometries.