An approximate semimolecular theory has been developed to investigate the composition dependence of Stokes shift dynamics of a fluorescent dye molecule dissolved in binary mixtures of an ionic liquid (IL) with a conventional polar solvent at different mole fractions. The theory expresses the dynamic Stokes shift as a sum of contributions from the dye IL and the dye polar solvent interactions and suggests substantial solute cation dipole dipole interaction contribution to the solvation energy relaxation. The theory, when applied to aqueous mixtures of 1-butyl-3-methylimidazolium hexafluorophosphate ([Bmim][PF6]) and tetrafluoroborate ([Bmim] [BF4]), and binary mixtures of ([Bmim][BF4] + acetonitrile), predicts reduction of Stokes shift but acceleration of the dynamics upon increasing the polar solvent concentration for the most part of the mixture composition. The decrease in dynamic Stokes shift values has been found to occur due to decrease of the dye IL interaction in the presence of the added polar solvent. For aqueous binary mixtures of IL, the predicted results are in semiquantitative agreement with the available experimental results. However, the calculated dynamics suggest much weaker composition dependence than that observed in experiments. In addition, the theory predicts a turn around for dynamic Stokes shift in its composition dependence for ([Bmim][BF4] + acetonitrile) mixtures at higher dilutions of the IL. Interestingly, effective dipolar medium calculations for Stokes shift dynamics in ([Bmim][BR4] + dichloromethane) binary mixtures predict a very weak or even nonexistent nonlinear composition dependence. These predictions should be reexamined in experiments.