A new cyclic model of a class of chemical engine-driven chemical pumps, which operate among three reservoirs at different chemical potentials, is set up. The influence of the irreversibility of finite-rate mass transfer between the cyclic working fluid and the three reservoirs on the performance of the chemical pumps is taken into account. The optimal chemical potentials of the cyclic working fluid and the optimal relation between the coefficient of performance and the rate of energy pumping for the chemical pumps are derived. On the basis of the optimal relation, some other optimal performances of the chemical pumps are discussed. For example, the maximum rate of energy pumping and the corresponding coefficient of performance, the optimal mass-transfer time, the minimum rate of entropy production, and so on, are calculated and their general expressions are given. The results obtained here cannot only enrich the theory of thermodynamics but also provide some theoretical guidance for the optimal design and development of a class of chemical engine-driven chemical pumps. Moreover, some important conclusions relative to the chemical pumps operating between two reservoirs at different chemical potentials, which have been investigated in the literature, can be deduced directly from the results in this paper.