What can be said about the Fourier expansion of compositions of Boolean functions, if anything? Let $\chi_S: \mathbb{F}_2^n \to \{\pm 1\}, x \mapsto (-1)^{\sum_{i \in S}x_i}$ for every $S \subseteq [n]$ denote the parity functions from $\mathbb{F}_2^n$ to $\{\pm 1\}$. Then, of course, every $f: \mathbb{F}_2^n \to \{\pm 1\}$ can be written as a linear combination (over $\mathbb{R}$) of the parity functions. I have tried (without much ease or success), to produce the Fourier expansions of compositions of even simple Boolean functions, in particular of conjunctions of parity functions, like $\chi_S \wedge \chi_T$ (where under the encoding of $1_{\mathbb{F}_2^n} \mapsto -1,0_{\mathbb{F}_2^n} \mapsto +1$, $\wedge$ is just $\max_2$).
So, my questions are:

Are there good techniques/tools in general for computing/estimating the Fourier expansion of $f \circ g$ when we know both of the expansions for both $f,g$?
Is there something we can say about the particular case of conjunctions/disjunctions over parity functions?
If $f,g$ are two $\pm1$-valued functions and $f \lor g$ is their maximum (this is the standard notation for maximum; $\land$ stands for minimum), then
$$
f \lor g = \frac{1 + f + g - fg}{2}.
$$
Since $\chi_S \chi_T = \chi_{S \Delta T}$, this shows that
$$
\chi_S \lor \chi_T = \frac{1}{2} + \frac{\chi_S}{2} + \frac{\chi_T}{2} - \frac{\chi_{S\Delta T}}{2}.
$$
You can read the Fourier expansion from this (unless $S = T$, in which case $\chi_S \lor \chi_S = \chi_S$).