The dynamics of ions and solvent molecules in polyelectrolyte desalination membranes is key to water purification technologies in which selective transport of the different components is desired. Recent experimental and our computational results have shown that nontrivial mechanisms underlie the transport properties of salt ions and water in charged polymer membranes. Explicitly, in polymer electrolytes, we found a reversal in the salt concentration dependence of the mobilities of Na+, Cl- salt ions and water molecules when compared with aqueous solutions. Motivated by such results, in this study, we have used atomistic molecular dynamics simulations to probe whether the mechanisms deduced in our earlier work apply to other salt systems and to mixtures of salts. Specifically, we report results for the ion diffusivities in aqueous KCl, MgCl2, and a 1:1 mixture of NaCl and MgCl2 salt solutions at different concentrations (ranging from 0.06 to 1 M) and investigate, at the molecular level, the mechanisms underlying the behaviors of salt and water transport properties. Our results show that diffusion of salt ions and water in charged polymer membranes are in general influenced by their association with polymer charge groups and ion pairing effects. Divalent ions are more strongly coupled with the polymeric ionic groups than monovalent salt ions and exhibit diffusivity trends that are distinct relative to monovalent salts. Further, we demonstrate that the mobilities of water molecules are influenced by coordination of water with polymer charge groups and their ion pairing tendencies and also exhibit distinct trends in monovalent and divalent salt solutions.