This paper analyses the mechanisms of ion transport and storage in insertion materials such as metal-oxide electrochromics, Li ion batteries and conducting polymers. We generalize the normal description of intercalation kinetics, based on diffusion of a lattice gas with mean field interactions, using an intercalation model system composed of different types of intercalation sites: a shallow site that allows diffusion of intercalated ions throughout the film, and deeper sites that trap the inserted charge for long times. The trapping is related to ion binding processes, like Li+ forming a bond to a bridging-type oxygen in a solid, or ion binding in micro-cavities of a polymer matrix. A complete classification of patterns of spectra for ion diffusion and trapping (complex impedance and complex capacitance) is provided, in terms of characteristic frequencies that depend on the steady state. We discuss the possible origin of the variation of the chemical diffusion coefficient with the composition. In a two state system the effective diffusion coefficient increases by several orders of magnitude as the chemical potential μ passes through the lower energy level, filling the trap states. We also show that when the interactions are treated correctly, the variations of diffusivity in a lattice gas are larger than previously expected.