Suppose I am given $n$ samples of sizes $N_1, \dots, N_n$ from a Dirichlet–multinomial distribution: Fixed and given is a $k$-vector $\mathbf{\alpha}$ of positive real numbers. For each $i, \, 1 \le i \le n$, a random probability vector $\mathbf{p}_i$ is drawn from a Dirichlet distribution $\mathrm{Dir}(\mathbf{\alpha})$, and then a sample of size $N_i$ is drawn from a multinomial distribution on $\{1, \dots, k\}$ with probabilities given by $\mathbf{p}_i$. The observed frequencies are recorded in an $n \times k$ table. Assume that there are no problems with low cell counts.
What can one say about the value $X$ of the $\chi^2$-statistic that can be computed for this random table? We can always write $\mathbf{\alpha} = M \mathbf{p_o}$ for some probability vector $\mathbf{p_o}$, and if $M$ is large, then all rows in this random table come from approximately the same multinomial distribution, so approximately $X \sim \chi^2_{r}$ with $r = (n-1)(k-1)$. How large must $M$ be for this to happen? And what happens for small $M$?
