  Let $(\xi_k,\eta_k)_{k\in\mathbb{N}}$ be independent identically distributed
random vectors with arbitrarily dependent positive components. We call a
(globally) perturbed random walk a random sequence $T:=(T_k)_{k\in\mathbb{N}}$
defined by $T_k:=\xi_1+\ldots+\xi_{k-1}+\eta_k$ for $k\in\mathbb{N}$. Consider
a general branching process generated by $T$ and denote by $N_j(t)$ the number
of the $j$th generation individuals with birth times $\leq t$. We treat early
generations, that is, fixed generations $j$ which do not depend on $t$. In this
setting we prove counterparts for $\mathbb{E}N_j$ of the Blackwell theorem and
the key renewal theorem, prove a strong law of large numbers for $N_j$, find
the first-order asymptotics for the variance of $N_j$. Also, we prove a
functional limit theorem for the vector-valued process $(N_1(ut),\ldots,
N_j(ut))_{u\geq 0}$, properly normalized and centered, as $t\to\infty$. The
limit is a vector-valued Gaussian process whose components are integrated
Brownian motions.
