What is the dynamic process of Compton scattering, and why is it postulated that the collision is an elastic one? What is the dynamic process of Compton scattering? Does it happen instantaneously? If not, what is the changing physical process?
And why is it postulated that the collision is an elastic one?
 A: The Compton scattering $\gamma e^- \to \gamma e^-$ can be regarded as an elastic process in the meaning that the total kinetic energy of the system is conserved (the final particles are the same as the initial ones).
However to comprehend the interaction at both low energy limit $\omega \lt \lt m$ and high energy limit $\omega \gt \gt m$, you require the QFT (quantum field theory) description. The perturbation approach computes the scattering matrix elements at different expansion orders in the coupling $e$. The interaction terms are pictorially featured by the Feynman diagrams.
In QED (quantum electrodynamics) at the $e^2$ order there are two diagrams that, according to the Mandelstam variables, are defined as s-channel, which is an annihilation process where the intermediate state (momentum squared in the propagator) is timelke, and as t-channel, which is a scattering process where the intermediate state is spacelike. As the cross section is related to the modulus squared of the sum of the diagrams, it is of the $e^4$ order.
The low energy limit gives the Klein-Nishina formula, that in the limit $m \to \infty$ reduces to the Thomson scattering cross section for classical electromagnetic radiation by a free electron. The high energy limit allows to understand the spin and the polarization dependence.
