Certain qualitative changes in turbulent eddy mixing during transitional regimes towards stronger stratification are highlighted using high-resolution numerical simulations. The behavior of turbulent eddy mixing parameters found in this study is consistent with some recent observational results in stably stratified shear flows in the atmosphere. In particular, the flux Richardson number increases, saturates and then decreases after a critical value of the gradient Richardson number (Rig) around 2.0. Correspondingly, there is also a transition from a decreasing trend in the turbulent eddy mixing coefficient for momentum to an increasing one at the same critical Rig. The implication of the study is that such a transition can be of quite general interest in sheared flows under stronger stratification (with significant wave dynamics), and needs to be accounted for in the parameterization of microscale atmospheric turbulence.