Given a partition $\lambda=(\lambda_1,\ldots,\lambda_r)$ of $n$, we have a Young permutation representation $M^\lambda$ of $S_n$. The linear representation $M^\lambda$ is the free vector space generated by $\Omega^\lambda$, the elements of which are partitions of $[n]=\{1,\ldots,n\}$ into labeled subsets $T_1,\ldots,T_r$ with $|T_i|=\lambda_i$. The defining action of $S^n$ on $[n]$ induces an action on $\Omega^\lambda$ and therefore on $M^\lambda$.
It is known how $M^\lambda$ decomposes into Specht modules -- that is, the irreducible representations of $S_n$. There exists one Specht module $S^\mu$ for each partition $\mu$ of $n$. We know that $S^\mu$ appears as a subrepresentation of $M^\lambda$ if and only if $\mu$ dominates $\lambda$ -- i.e. if $\sum_{i=1}^m \mu_i\ge\sum_{i=1}^m\lambda_i$ for each $m\in\mathbb{N}$. The multiplicity of $S^\mu$ in $M^\lambda$ is also known.
Here is my question. Consider the partition $\lambda$ as a multiset with elements $1,\ldots,r$ and where $i$ occurs with multiplicity $\lambda_i$. A bijection of multisets $\lambda\to\lambda'$ determines a morphism $M^\lambda\to M^{\lambda'}$. I would like a reference for what this morphism looks like on Specht submodules of $M^\lambda$. More precisely, for each isomorphism class of irreducible representation present in $M^\lambda$, the restriction of this morphism to that isomorphism class should be described by a matrix with number of columns equal to the multiplicity of the representation in $M^\lambda$ and the number of rows equal to the multiplicity in $M^{\lambda'}$.
I think I could work this out by examining the proof of the decomposition of $M^\lambda$ and with a bit of elbow grease, but I would rather not reinvent the wheel (and a short citation would be preferable to a longer proof), so I'm wondering if an explanation of this morphism appears anywhere in the literature.