  We consider graph classes $\mathcal G$ in which every graph has components in
a class $\mathcal{C}$ of connected graphs. We provide a framework for the
asymptotic study of $\lvert\mathcal{G}_{n,N}\rvert$, the number of graphs in
$\mathcal{G}$ with $n$ vertices and $N:=\lfloor\lambda n\rfloor$ components,
where $\lambda\in(0,1)$. Assuming that the number of graphs with $n$ vertices
in $\mathcal{C}$ satisfies \begin{align*} \lvert \mathcal{C}_n\rvert\sim b
n^{-(1+\alpha)}\rho^{-n}n!, \quad n\to \infty \end{align*} for some $b,\rho>0$
and $\alpha>1$ -- a property commonly encountered in graph enumeration -- we
show that \begin{align*} \lvert\mathcal{G}_{n,N}\rvert\sim c(\lambda)
n^{f(\lambda)} (\log n)^{g(\lambda)} \rho^{-n}h(\lambda)^{N}\frac{n!}{N!},
\quad n\to \infty \end{align*} for explicitly given
$c(\lambda),f(\lambda),g(\lambda)$ and $h(\lambda)$. These functions are
piecewise continuous with a discontinuity at a critical value $\lambda^{*}$,
which we also determine. The central idea in our approach is to sample objects
of $\cal G$ randomly by so-called Boltzmann generators in order to translate
enumerative problems to the analysis of iid random variables. By that we are
able to exploit local limit theorems and large deviation results well-known
from probability theory to prove our claims. The main results are formulated
for generic combinatorial classes satisfying the SET-construction.
