What is the basic lemma on composition of probability generating functions? What is the basic lemma on composition of probability generating functions and how is it most clearly proved?
(I'm posting this mainly to see if I can write an answer as clearly and simply as possible, but maybe other people know things about this that I don't and can post answers from their points of view.)

Do you mean this?  Let $S = \sum_{i\le N} Y_i$ where $N, Y_1, Y_2, \ldots$ are independent integer-valued random variables where $N\ge 0$ a.s., and $\mathbb E[z^N] = f(z)$ while $\mathbb E[z^{Y_i}] = g(z)$, then
$$ \mathbb E[z^S] = f(g(z))$$
