The Hecke algebras $\mathcal{H}_{n,k}$ of the group pairs $(S_{kn}, S_k\wr S_n)$ can be endowed with a filtration with respect to the orbit structures of the elements of $S_{kn}$ relative to the action of $S_{kn}$ on the set of $k$-partitions of $\{1,\dots,kn\}$. We prove that the structure constants of the associated filtered algebra $\mathcal{F}_{n,k} $ is independent of $n$. The stability property enables the construction of a universal algebra $\mathcal{F}$ to govern the algebras $\mathcal{F}_{n,k}$. We also prove that the structure constants of the algebras $\mathcal{H}_{n,k}$ are polynomials in $n$. For $k=2$, when the algebras $(\mathcal{F}_{n,2})_{n\in \mathbb{N}}$ are commutative, these results were obtained by Aker and Can, by Can and Ozden, and by Tout.