The application of liquid−liquid chromatography principles to solvent extraction processes in hydrometallurgy can greatly simplify rare earth metal separation technologies by separating multicomponent mixtures in one technological operation. In this study, the chromatographic separation of rare earth elements (REEs) in multistage extraction columns was experimentally studied under conditions of impulse sample injection—single and multiple loading of large volumes of metal salt solution into the installation. The results obtained showed the feasibility of operating sieve plate extraction columns in the liquid−liquid chromatography mode. A closed-loop recycling technology is proposed for the separation of rare earth elements in multistage extraction columns operating in the liquid−liquid chromatography mode. For further development and industrial implementation of this technology, experimental studies should be conducted on intensified multistage extraction columns, such as sectioned columns with agitators and vibrating plate columns. Computer simulation of the chromatographic separation of rare earth elements by closed-loop recycling liquid−liquid chromatography was carried out.