quantum field theory - Scattering, Perturbation and asymptotic states in LSZ reduction formula

I was following Schwarz's book on quantum field theory. There he defines the asymptotic momentum eigenstates $|i\rangle\equiv |k_1 k_2\rangle$ and $|f\rangle\equiv |k_3 k_4\rangle$ in the S-matrix element $\langle f|S|i\rangle$ as the eigenstates of the full Hamiltonian i.e., $H=H_0+H_{int}$. Therefore, the states $|i\rangle=|k_1 k_2\rangle$ is defined as

$$|k_1 k_2\rangle=a_{k_1}^{\dagger}(-\infty) a_{k_2}^{\dagger}(-\infty)|\Omega\rangle$$

where $|\Omega\rangle$ is the vacuum of the full interacting theory. Then the LSZ reduction formula connects the S-matrix element $\langle f|S|i\rangle$ to the Green' functions of the interaction theory defined as

$$G^{(n)}(x_1,x_2,...x_n)=\langle \Omega|T[\phi(x_1)\phi(x_2)...\phi(x_n)]|\Omega\rangle.$$ Here are a few doubts.

Doubt 1 When the particles are far away, the interaction can be considered to be adiabatically switched off. Therefore, at $t=\pm\infty$ the states are really free particle states and should have been written as

$$|k_1 k_2\rangle=a_{k_1}^{\dagger}(-\infty) a_{k_2}^{\dagger}(-\infty)|0\rangle$$

and

$$|k_3 k_4\rangle=a_{k_3}^{\dagger}(+\infty) a_{k_4}^{\dagger}(+\infty)|0\rangle$$ where $|0\rangle$ is the vacuum of the free thoery. I do not understand why these the states $|i\rangle$ and $|f\rangle$ are derived from $|\Omega\rangle$ instead of $|0\rangle$.

Doubt 2 The initial and final states were derived from the vacuum of the interacting theory $|\Omega\rangle$. According to my understanding, this suggests that the states $|i\rangle\equiv |k_1 k_2\rangle$ and $|f\rangle\equiv |k_3 k_4\rangle$ are eigenstates of the full Hamiltonian $H$. Since then there is no perturbation, there should not be any scattering or transition at all.

---

More references Even Peskin and Schroeder, Bjorken and Drell, Srednicki take the same approach as Schwartz; they too define the external momentum eigenstates to the eigenstate of the full Hamiltonian $H$. However, if the system was initially in a stationary state why should it undergo a transition in absence of any perturbation?