Suppose that you have two random variables $x\sim F_x$ and $y\sim F_y$ where $F_x,F_y$ are probability distributions. Then generally $w_xx+w_yy$ is not from $w_xF_x+w_yF_y$, even if when $F_x,F_y$ are normal distributions.
It is true that the characteristic function of the weighted sum is the sum of characteristic functions of these distributions: $\psi_{w_xx+w_yy}=w_x\psi_x+w_y\psi_y$. You can obtain the distribution $F_{w_xx+w_yy}$ from its characteristic function, of course.
Finally, the whole idea of the Gaussian mixture is to model unknown distributions, such as those with fat tails in finance.