How to obtain a Dirichlet-multinomial distribution?

Question

Let $X_i, i=1, \ldots, n$ be independent Poisson random variable with parameters $\lambda_i$ correspondingly and conditioned such that $\sum_{i=1}^nX_i=A$.

It is known that $(X_1,\ldots,X_n)$ conditioned on $\sum_{i=1}^n X_i = A$ is multinomial with $p_i = \lambda_i/\sum_j \lambda_j$, $i=1,\ldots,n$, and number of trials $A$.

In the paper https://arxiv.org/pdf/2001.04343.pdf stated that multinomial distribution is related to Dirichlet-multinomial distribution.

1. What is the relation between these two distributions?
2. Let $a_i, i=1, \ldots, n \in R$. Denote $S=\sum_{i=1}^N a_iX_i$. Can $E|S|^p, p\geq2$ be computed/estimated through Dirichlet-multinomial distribution?