Quantum Mechanical amplitude for particle falling into a black hole

Consider a black hole spacetime originated by gravitational collapse, like the following Vaidya geometry

$$ds^2=-\left(1-\frac{2M\theta(v)}{r}\right)dv^2+2dvdr+r^2(d\theta^2+\sin^2\theta d\phi^2).$$

Next, consider a massless Klein-Gordon field propagating in said background. Consider the asymptotic quantization on $\mathcal{I}^-$. We can introduce a set of solutions $f_i$ which are positive frequency with respect to advanced time and are orthonormal in the KG inner product. This decomposes the field as $$\phi=\sum_i a_i f_i+a_i^\dagger f_i^\ast.$$

This defines a vacuum $a_i|0\rangle_{\text{in}}=0$ and one associated (in) Fock space $\mathscr{F}_{\text{in}}$. One such state is a state on which we have a number of ingoing massless particles.

Next suppose the field is on some one-particle state $$a_i^\dagger|0\rangle=|1_i\rangle.$$ In other words: an observer in the asymptotic past Minkowskian region will see one particle in some mode.

Now, the particle enters spacetime and interact only gravitationaly (in other words, the field doesn't interact with anything else apart from gravity).

The question is: how can I mathematically describe quantum mechanically the amplitude for the particle fall through the horizon into the black hole?

I don't see how to do this because I'm used to getting amplitudes out of observables and eigenstates of such observables. So if $A$ is an observable with eigenstates $|a_i\rangle$ the probability amplitude for result $i$ is $\langle a_i|\phi\rangle$ on state $|\phi\rangle$. This doesn't seem to fit here.

Moreover an associated and important question is: suppose that we have two particles $$a_i^\dagger a_j^\dagger |0\rangle = |1_i, 1_j\rangle,$$ now suppose I want to consider that I want to compute things for one particle conditional to the other having fallen into the black hole.

Again in usual QM I would have an observable, consider a measurement and take the post-selected state associated to a measurement. Here I don't know how this could be done.

Are perhaps these questions totally ill-posed? In that sense how would quantum-mechanically we talk about quantum particles falling into black holes? After all this seems to be something quite reasonable to happen.

Edit: this question of mine is related. Some authors like Parker and Hawking claim that a certain quantity is the "absorption cross section for a mode of frequency $\omega$". In particular, I believe that the right answer to this question should in the end connect to that and yield the absorption cross section these authors mention.