Let $\pi,\sigma\in S_n$ be two permutations, and $\sigma\circ\pi\in S_n$ denotes their composite permutation: $$\sigma\circ\pi=[\sigma(\pi(1)), ...,\sigma(\pi(n))]$$ which can also be seen as relabling elements of $\pi$ according to $\sigma$. Moroever, let $\mathcal{I}\colon S_n\to\mathbb{N}$ be a function that counts inversions in a permutation: $$\mathcal{I}(\pi)=\sum_{i<j}\mathbb{1}_{\{\pi(i)>\pi(j)\}}$$ And finally let distance between the two permutations be defined as: $$d(\pi_1,\pi_2)=\#\{\pi_1(i)<\pi_1(j):\pi_2(i)>\pi_2(j)\}$$
Now suppose we have two permutations $\pi_1,\pi_2\in S_n$ and $\sigma$ is chosen uniformely at random $\sigma\sim\text{Unif}(S_n)$.
What is the distribution of $\mathcal{D}:=\mathcal{I}(\sigma\circ\pi_1)-\mathcal{I}(\sigma\circ\pi_2)$? Can we compute its expected value $\mathbb{E}[\mathcal{D}]$ higher moments as a function of $d(\pi_1,\pi_2)$? In other words, what is the concentration of $\mathcal{D}$ around its mean? If $d(\pi_1,\pi_2)$ does not provide enough information about $\mathcal{D}$ is there some other naturally defined distance that is more relevant?
As it's been pointed out in the comments, $E[\mathcal{D}]=0$, now the question is what are the higher moments of $\mathcal{D}$, for instance, what is $Var[\mathcal{D}]$? and its higher moments?