We analyse the approach towards local isotropy in statistically stationary turbulent shear flows using the transport equations for the fourth-order moments of the velocity derivative. It is found that terms of these equations representing the large-scale contribution associated with the uniform mean velocity gradient gradually decrease as the Taylor microscale Reynolds number $Re_\lambda$ increases, and finally disappear when $Re_\lambda$ is sufficiently large. This gradual weakening of the large-scale effect is accompanied by a gradual approach towards local isotropy of the small-scale motion. The rate at which local isotropy is approached depends on the weakening of the large-scale forcing, which is controlled by the magnitude of the non-dimensional velocity shear parameter $S^*$ ($\equiv \overline {u_1^2}({{\partial {{\bar U}_1}}}/{{\partial {x_2}}})/{\bar {\varepsilon }_{iso}}$, where $\bar {\varepsilon }_{iso}$ is the isotropic mean turbulent energy dissipation rate, $\overline {u_1^2}$ is the streamwise velocity variance, and ${\partial {{\bar U}_1}/\partial {x_2}}$ is the uniform mean velocity gradient in the transverse direction). In particular, we show that the approach towards local isotropy can be recast in the form $C\, Re_\lambda ^{-1}$, where $C$ is the product of $S^*$ and a ratio of transverse-to-streamwise velocity derivative variances. This is consistent with the behaviour of the normalized third-order moments of transverse velocity derivatives. With the further use of the transport equations for the eighth- and twelfth-order velocity derivative moments, it is found that the even moments of transverse velocity derivatives can significantly affect the rate at which local isotropy is approached, especially for higher orders.