The possibility of the formation of an alkali–silicate reaction (ASR) is a crucial issue for the service life of concrete. The coexistence of key parameters such as the presence of alkalis, reactive SiO2, humidity, and temperature predetermine the possibility of its formation and application. When an ASR gel forms, it results in the concreting cracking and spalling as well as in the deterioration of its overall properties. The risk of ASR depends on the concentration of alkalis and their mobility, which influence their ability to penetrate the concrete. The objective of this study was to determine the ionic mobility of not only Na+ and K+, but Ca2+ as well, from external sources (0.5 and 1.0 mol/L solutions of Na/K carbonate, nitrate, and hydroxide) to a cementitious matrix as the precursor for ASR. The concentrations of ions in both the immersion solutions (ICP) and the cementitious matrix itself (SEM-EDX) were studied as a function of time, from 0 to 120 days, for leaching, and according to temperature (25 and 40 °C). The reaction products were characterized using SEM-EDX. Different diffusion rates and behavior were observed depending on the anion type of the external alkali source. Both sodium and potassium ions in all the three environments studied, namely carbonate, hydroxide, and nitrate, penetrated into the composite and further into its structure by different mechanisms. The action of hydroxides, in particular, transformed the original hydration products into calcium-silicate-hydrate (CASH) or ASR gel, while nitrates crystallized in pores and did not cause any changes in the hydration product. The driving force was the increased temperature of the experiment as well as the increased concentration of the solution to which the test specimen was exposed.