Suppose we have a formal power series $h(t)=t+\sum_{k=2}^\infty h_k(t^k/k!)$. In principle, this can be inverted to obtain $g(x)=x+\sum_{k=2}^\infty g_k(x^k/k!)$ such that $h(g(x))=x$. The specific expressions are known in terms of Bell polynomials and can be found on Wikipedia's page on the Lagrange inversion theorem.
Now, suppose we have another formal power series $y(t)=t+\sum_{k=1}^\infty y_k (t^k/k!)$ with $y_1=1$. Then we can eliminate the $t$-dependence via the above reversion of series, yielding another formal power series $f(x)=(y\circ g)(x)=x+\sum_{k=2}^\infty f_k (x^k/k!)$. Are the expressions for the coefficients $\{f_k\}$, in terms of $\{y_k\}$ and $\{x_k\}$, known and documented somewhere?
An elegant method is to use the Expansion Theorem in Umbral Calculus. Below is a typical statement, from p. 18 of Roman's book The Umbral Calculus.