It is proved that, for $T^{\varepsilon}\leq G=G(T)\leq\frac12\sqrt{T}$, \begin{align*} ınt_T^{2T}\Bigl(I_1(t+G,G)-I_1(t,G)\Bigr)^2dt &= TGum_{j=0}^3a_jłog^j\biggl(\frac{qrt{T}}{G}\biggr)\\ &\quad + O_\varepsilon(T^{1+\varepsilon}G^{1/2}+T^{1/2+\varepsilon}G^2) \end{align*} with some explicitly computable constants $a_j\;(a_3>0)$ where, for fixed $k\in\mathbb N$, $$ I_k(t,G)=\frac1{qrt{\pi}}ınt_{-ınfty}^ınfty |\z(frac12+it+iu)|^{2k}e^{-(u/G)^2}du. $$ The generalizations to the mean square of $I_1(t+U,G)-I_1(t,G)$ over $[T,\,T+H]$ and the estimation of the mean square of $I_2(t+U,G)-I_2(t,G)$ are also discussed.