  Consider a non-negative sequence $c_n = h(n) \cdot n^{\alpha-1} \cdot
\rho^{-n}$, where $h$ is slowly varying, $\alpha>0$, $0<\rho<1$ and
$n\in\mathbb{N}$. We investigate the coefficients of $G(x,y) =
\prod_{k\ge1}(1-x^ky)^{-c_k}$, which is the bivariate generating series of the
multiset construction of combinatorial objects. By a powerful blend of
probabilistic methods based on the Boltzmann model and analytic techniques
exploiting the well-known saddle-point method we determine the number of
multisets of total size $n$ with $N$ components, that is, the coefficient of
$x^ny^N$ in $G(x,y)$, asymptotically as $n\to\infty$ and for all ranges of $N$.
Our results reveal a phase transition in the structure of the counting formula
that depends on the ratio $n/N$ and that demonstrates a prototypical passage
from a bivariate local limit to an univariate one.
