The question relates to the numbers $H(n, m)$ of the Anand–Dumir–Gupta conjectures solved by Richard Stanley, see e.g. [1] (this MO post set me on this track).
Indeed, we have:
Claim 1. Let $M$ be a multiset of size $mn$ consisting of $n$ elements $a_1, \dots, a_n$, each with multiplicity $m$. Let $X = (x_s)_{1 \le s \le mn}$ with $x_s \in \{a_1, \dots, a_n\}$ be a permutation of $M$ and let $\mu = (\mu_{i, j})_{1 \le i, j \le n} \in \mathbb{Z}_{\ge 0}^{n \times n}$ be the matrix defined $$\mu_{i, j} = \#\left\{s \, \vert \, x_s = a_j,\, im + 1 \le s \le (i + 1)m\right\}.$$ Then the sum of every row and every column of $\mu$ is $m$.
Proof. A straightforward verification.
Note. The integer $H(n, m)$ of the Anand–Dumir–Gupta conjectures is the number of matrices in $\mathbb{Z}_{\ge 0}^{n \times n}$ with constant row and column sums equal to $m$, see [1].
We shall establish:
Claim 2. Let $M$ be a multiset as in Claim 1. Any two permutations of $M$ can be related by means of at most $m(n - 1)$ swaps.
Our proof relies on the following lemma:
Lemma 1 (Birkhoff-von Neumann, see [1, Theorem 1.1]). Let $\mu \in \mathbb{Z}_{\ge 0}^{n \times n}$ be a matrix such that the sum of every of its rows and of every of its columns is $m$. Then $\mu$ is a sum of $m$ permutation matrices.
Our strategy is rather naive. It consists in showing that a permutation of $M$ decomposes as a union of $m$ intertwined permutations of $\{a_1, \dots, a_n\}$.
This is the purpose of the next lemma.
Lemma 2. Let $M$ and $X$ be as in Claim 1. Then there are pairwise distinct indices $i_{k, l} \in \{1, \dots, mn\}$ with $k \in \{1, \dots, m\}$ and $l \in \{1, \dots, n \}$ such that
- $i_{k, l} \in [(l - 1)m + 1, lm]$ for every $k$ and every $l$,
- $\{x_{i_{k,l}} \, \vert \, 1 \le l \le n\} = \{a_1, \dots, a_n\}$ for every $k$, i.e., the map $\sigma_k$ defined by $$\sigma_k(a_l) = x_{i_{k, l}}$$ is a permutation of the set $\{a_1, \dots, a_n\}$ for every $k$.
Proof. Apply Claim 1 and Lemma 1: the matrix $\mu = \mu(X)$ is a sum of $m$ permutation matrices. The correspondings permutations are the desired permutations $\sigma_k$.
We are now in position to prove Claim 2.
Proof of Claim 2. Each of the $m$ permutations $\sigma_k$ of Lemma 2 can be turned into the identity by means of at most $n - 1$ swaps ($n - 1$ is the maximal reflection length of the permutations of $\{a_1, \dots, a_n\}$). We conclude by observing that the minimum number of swaps $w(X, Y)$ required to turn $X$ into $Y$, for $X$ and $Y$ two permutations of $M$, satisfies: $$w(X \sigma, Y \sigma) = w(X, Y)$$ for every $\sigma \in \text{Sym}(mn)$.
It is natural to ask:
Question. Is the upper bound of Claim 2 a tight bound? More specifically, does $w(m,n) \ge m(n - 1)$ hold?
W. Bruns, "Commutative Algebra Arising from the Anand–Dumir–Gupta Conjectures", 2007.