Expected squared distance between order statistics? Suppose $p(\cdot)$ is a smooth probability distribution over $\mathbb R$.  Suppose we draw two collections of $k$ i.i.d. samples from $p(\cdot)$, yielding random variables $(X_1,\ldots,X_k)$ and $(Y_1,\ldots,Y_k)$.  Use $X_{(i)}, Y_{(i)}$ to denote the $i$-th order statistic of $(X_1,\ldots,X_k), (Y_1,\ldots,Y_k)$, resp.; that is, $X_{(i)}$ is the $i$-th value obtained when sorting the sample $X_1,\ldots,X_k$.
Is it possible to compute in closed-form the following expectation?
$$
\mathbb E_{\substack{X_1,\ldots,X_k\sim p(\cdot)\\Y_1,\ldots,Y_k\sim p(\cdot)}}\left[
\sum_{i=1}^k (X_{(i)}-Y_{(i)})^2
\right]
$$
That is, is there an expression for the expected squared L2 distance between $\vec X$ and $\vec Y$ after sorting?
This answer gets part of the way, showing roughly that the expected value tends to zero as $k\to\infty$.  This document (and many others) gives the distribution function for each $X_{(i)}$.  If this value isn't known in closed-form, anything about its properties as a function of $k$ would be helpful as well!
 A: The terms of this sum are known exactly for some distributions, since
$$E[(X_{(i)}-Y_{(i)})^2]=E[X_{(i)}^2]-2E[X_{(i)}]E[Y_{(i)}]+E[Y_{(i)}^2]=2\text{Var}[X_{(i)}].$$
For those distributions, we can get some nice expressions for the limiting behavior.
In what follows, $\psi_1(z)$ is the trigamma function $d^2\log \Gamma(z)/dz^2$, appearing in the calculation for the logistic distribution in N. Balakrishnan and A. Clifford Cohen, Order Statistics and Inference (1991), p. 40. The approximations given here with $\sim$ are exact in the highest-order terms.
For a uniform distribution bounded by $0,1$:
\begin{align}
\text{Var}[X_{(i:k)}]&=\frac{i(k+1-i)}{(k+1)^2(k+2)}\phantom{+123456}\\
\\
2\sum_i\text{Var}[X_{(i:k)}]&=\frac{k}{3(k+1)}\sim\frac13
\end{align}
For an exponential distribution with mean $1$:
\begin{align}
\text{Var}[X_{(i:k)}]&=\psi_1(k+1-i)-\psi_1(k+1)\\
\\
2\sum_i\text{Var}[X_{(i:k)}]&\sim 2\log k+1
\end{align}
For a logistic distribution with parameters 0,1:
\begin{align}
\text{Var}[X_{(i:k)}]&=\psi_1(k+1-i)+\psi_1(k)\phantom{+12}\\
\\
2\sum_i\text{Var}[X_{(i:k)}]&\sim 4\log k+8
\end{align}
For the unbounded distributions, these are growing functions: the answer linked in the question shows $\frac{1}{k}\|X-Y\|_1\to 0$, while this is showing $\|X-Y\|_2^2\to \infty$.
