I believe they are both correct. The difference between them is how time is treated. "over the course of" emphasizes that during this designated period of time, a thing occurs. Elapsed time and stated occurrence/s on a whole is the intended message. In contrast, "through the course of", particularly emphasizes a more real-time, perhaps thorough, and/or minutiae-based treatment of said occurrence/s during said designated period of time.
I think the difference in meaning and typical usage is due possibly to the connotations/implications of each phrases' respective prepositions: "over" & " through". Contrasting the two, the preposition "over" could more easily be interpreted as clearing, i.e. being/going/moving above, not or no longer having to interact with nor be impeded by X to same degree as when level with it, whereas the preposition "through" may often involve having to navigate one or a myriad of objects, areas, situations, etc, thereby perhaps requiring active focus, attention, and care in order to successfully complete the action; thus, resulting in the effect of being taken into a real-time continuous framework, allowing for detail and elaboration beyond the general, if you choose to do so, since you are going "through" it and not "over" it.
With that said, you don't have to add anything extra or nuanced when using "through the course of" as opposed to "over the course of", it's really just flavoring.
I hope that wasn't too convoluted; I had to parse it out in my head, but I think I'm in the ballpark.