Ah I realize the notation is confusing, sorry. The martingale is in $t$, the filtration is $\sigma(Y_{t-1},\dots,Y_0)$ where $Y_0 = n$ (equivalently $\sigma(M_{t-1},\dots,M_0))$. $n$ is a parameter. In words, $n$ is the number of people alive initially, at each ``round'' $t=1,2,\dots$ each person can die with probability $p$. $Y_t$ is the number of people alive at round $t$.

These are good points. Some interesting examples are found by taking a dirac delta $\delta_{x^n}$, and averaging it over all permutation. Convex combinations of such averages preserve $\|X^n\|^2 = n$. And indeed we can't recover a distribution of this form by Dominik's methods. I'm interested in marginals since I'm interested in which distributions $P_Y$ can be induced through a transform $P_{Y|X}$ of the marginal. Specifically, I'm maximizing some function of $P_Y$ over such distributions. But I thought the question of which distributions could be coupled is interesting alone.