This work is devoted to an extension of the continuum elasticity model combined with the classical lattice-gas (LG) model that recently appeared in the literature and described guest ion insertion into various host materials. The extension involves two important aspects: (i) derivation of intercalation isotherms with four different elastic parameters with the purpose to find their critical combination, leading to first-order phase transitions, and (ii) quantitative treatment of the first-order phase transition reactions controlled by slow external kinetics. This analysis helps to understand properly an important issue of the mechanism of electrochemically driven first-order phase transitions in various ion-insertion electrodes often disregarded in the literature: whatever the concerned electrochemical characteristics are measured, the underlying (often hidden, in practice) kinetic limitations must be considered. The analysis described herein refers to the dependence of the (chemical) differential intercalation capacitance, C dif, and of the chemical diffusion coefficient, D, on the concentration of Li-ions in graphite in the course of electrochemical lithiation/delithiation of these electrodes.