Let $f(x,t)\in\mathbb{C}[t][x]$ be monic of degree $d$ in $x$, separable over $\mathbb{C}(t)$, and irreducible in $\mathbb{C}(t)[x]$. Let $B\subset\mathbb{P}^1(\mathbb{C})$ be the finite branch (discriminant) locus and set $$ U=\mathbb{P}^1(\mathbb{C})\setminus B. $$ For $t\in U$, the equation $f(x,t)=0$ has $d$ distinct roots, giving a $d$-sheeted covering $$ X|_U(\mathbb{C})\to U(\mathbb{C}), $$ and hence a monodromy action of $\pi_1(U,t_0)$ on the $d$ roots, yielding a subgroup of $S_d$.
It is often stated that the Galois group of the splitting field of $f$ over $\mathbb{C}(t)$, $$ \mathrm{Gal}(f/\mathbb{C}(t)), $$ is equal (or naturally isomorphic) to the monodromy group.
What is a clean proof of this statement, with precise hypotheses? In particular, which covering's monodromy is meant (the original $d$-sheeted root cover, or a Galois-closure / ordered-roots cover), and what is the rigorous justification that a monodromy permutation induces a $\mathbb{C}(t)$-automorphism of the splitting field (rather than merely permuting generators)?