  Consider a general branching process, a.k.a. Crump-Mode-Jagers process,
generated by a perturbed random walk $\eta_1$, $\xi_1+\eta_2$,
$\xi_1+\xi_2+\eta_3,\ldots$. Here, $(\xi_1,\eta_1)$, $(\xi_2, \eta_2),\ldots$
are independent identically distributed random vectors with arbitrarily
dependent positive components. Denote by $N_j(t)$ the number of the $j$th
generation individuals with birth times $\leq t$. Assume that $j=j(t)\to\infty$
and $j(t)=o(t^a)$ as $t\to\infty$ for some explicitly given $a>0$ (to be
specified in the paper). The corresponding $j$th generation belongs to the set
of intermediate generations. We provide sufficient conditions under which
finite-dimensional distributions of the process $(N_{\lfloor
j(t)u\rfloor}(t))_{u>0}$, properly normalized and centered, converge weakly to
those of an integral functional of a stable L\'{e}vy process with finite mean.
