Let $(X_n)$ a sequence of independent real random variables. Let $N_n = \sum_{k=1}^n \delta_{X_i}$ its point process and an interval $I\subset \mathbb{R}$. Then, $$N_n(I)= \sum_{k=1}^n \delta_{X_i}(I) = \#{\{ k\leq n,X_k \in I \}}. $$ In other words, it is a discrete andom variable counting the number of $X_k$ in $I$. When $(X_n)$ is iid, $N_n(I)$ has Binomial law and when we only keep independence, $N_n(I)$ has Poisson-Binomial law.
Now consider the random vector $(N_n(I_1),...,N_n(I_p))$. In the same spirit, it think that when $(X_n)$ is iid, it has Multinomial law; but what about when when we only keep independence ?