A small eddy viscosity or mass diffusivity that varies with height has been found to have unexpected effects on the Kelvin-Helmholtz (KH) instability of a stably stratified shear layer near the neutral stability boundary. In particular, varying viscosity can increase the growth rate of the instability in contrast to the effect of uniform viscosity. Here, these results are extended to parameter ranges relevant in many geophysical and engineering contexts. We find that linearization of the viscous terms based on the assumption of weak viscosity/diffusivity is valid for nondimensional values (inverse Reynolds number) up to ~10⁻². Decreasing the Richardson number far below its critical value 1/4 can change, or even reverse, the effects of eddy viscosity and diffusivity. A primary goal is to explain the unexpected destabilization by viscosity. Varying viscosity affects vorticity (and other fluid properties) in a manner identical to advection with an advecting velocity equal to minus the gradient of viscosity. Destabilization occurs when this viscous “advection” reinforces the vorticity distribution of a growing mode.