What happens in the interval between the initial and final states of the interaction process?

What happens in between is everything and nothing. There is no privileged clearcut answer what happened that would be physically meaningful. It's really the very basic point of quantum mechanics that only results of measurements are physically meaningful facts or observables; all other data are fictitious or uncertain. By the very definition of your problem, no measurement took place in the intermediate states which means that no sharp answers to any questions were generated, no answers or values became real or privileged or facts.

Feynman's path integral formalism is the most explicit method to answer the question "what happened in between". In this approach, the only physically meaningful answer involves the summing over all possible intermediate histories that are weighted by $exp(iS/hbar)$ where $S$ is the action of each history. So what happens is the complex superposition of all conceivable intermediate histories with the given initial and final conditions.

Not only that. The absolute value of this exponential is always the same (namely one in my normalization) so all histories, whether they are close to an intuitive or classically allowed history, contribute equally. The nearly classical histories are favored in the classical limit due to the positive interference. The phase $S$ (an angle) is almost constant near the minimum of the action (note that the classically allowed history have the stationary or minimal action) and that is why their contribution to the final observed results is greatest. In a combination, these nearly classical histories contribute more than other combinations where much of the interference is destructive.

It's important to notice that the intermediate histories are indeed superpositions of qualitatively different histories. This point is made very explicit by the Feynman diagrams. The probability of a given processes is calculated from a sum of Feynman diagrams, each of which may have a different shape or even the number of virtual particles. All these diagrams contribute so "all the corresponding histories had some likelihood to happen" in between. But unlike classical physics, quantum mechanics says that not only the probabilities of each history matter. All the relative phases matter, too. As I said, most of the decisions "what looks real according to a quantum mechanical theory" boils down to the question whether the interference between the contributions is constructive or destructive.