For every set of natural numbers $A$ and for all positive integers $n$, $k$, let $c_k^A(n)$ be the number of compositions of $n$ into $k$ parts from $A$, that is, the number of $(a_1, \dots, a_k) \in A^k$ such that $a_1 + \cdots + a_k = n$.

I'm looking for asymptotic formulas for $c_k^A(n)$ as $n \to +\infty$ and $k \to +\infty$ (of course, in order to say anything interesting, some hypotheses on the relative grown of $n$ and $k$, and on the set $A$, are needed).

Where can I find such kind of results? I guess that something is surely known, but I keep finding only results with $A = \mathbb{N}$ and/or $k$ fixed.

Thank for any suggestion