I am struggling with the following quantile problem. Let $F_1$ and $F_2$ be the cumulative distribution functions, and let $F_1$ have its inverse denoted by $F_1^{-1}$. Let both functions be Hadamard differentiable and continuous, and let they have corresponding empirical measures $\hat{F}_1$, and $\hat{F}_2$. Assume also that the relation below is evaluated under proper domains. What is the difference between asymptotic distribution of $\hat{F}^{-1}_1(F_2)$ and $F^{-1}_1 (\hat{F}_2)$.
From the slides we have that $\hat{F}^{-1} = \phi(\hat{F})$ and by the functional delta method we have $$ \sqrt{n}(\phi(\hat{F}) - \phi(F)) = \phi_F'(\sqrt{n}(\hat{F} - F)) + o_P(1) = \phi_F'(\mathbb{G}_\lambda) + o_P(1), $$ where $\mathbb{G}_\lambda$ is the standard Brownian bridge. By distributive property of distribution functions and by the continuous mapping theorem we have that $\hat{F}^{-1}_1(F_2)$ converges weakly as
$$ \sqrt{n}(\hat{F}^{-1}_1(F_2) - F^{-1}_1(F_2)) = \sqrt{n}(\hat{F}^{-1}_1 - F^{-1}_1) \circ F_2 = \phi_{F_1}'(\mathbb{G}_{F_2}) + o_P(1), $$ where $\mathbb{G}_{F_2}$ is the $F_2$-Brownian bridge. Turning to $F^{-1}_1 (\hat{F}_2)$, the functional delta method should be used for function $F_1^{-1}$, i.e. $$ \sqrt{n}(F^{-1}_1(\hat{F}_2) - F^{-1}_1(F_2)) = (F^{-1}_1)_{F_2}'(\sqrt{n}(\hat{F}_2 - F_2) + o_P(1). $$ Now, I don't see how to derive the functional $(F^{-1}_1)_{F_2}'$ and in what dimension it is different from $\phi_{F_1}'$. For the latter we have (also from the slides) $$ \phi'_{F_1} (h) = - \left(\frac{h}{f_1} \right) \circ F_1^{-1} $$