The histories of Kush and Egypt are intertwined with each other because they were reliant on each other. How did their interdependence work, and why did each take control of the other in turn?

The Kingdom of Kush was located in Nubia, between Egypt and Sudan.

In simple terms, Egypt relied on Nubia's gold and Nubia relied on Egypt's grain. Each had something that the other could not produce by themselves, that's why they were willing to be interdependent. So much so, that historians to this day have tried to understand the entire sociopolitical structure of Kush through egyptian models, which is somewhat of a mistake as the Nubian state presented distinct cultural conditions since its earliest periods. We could say they were strongly reliant on each other economically, but Egypt has sort of taken control of the history of Kush.

Hope this clears things up!

How did their interdependence work (question 1)

Egypt limited the south with Kush or Nubia, where today the region of El Sudan is located. Egyptian texts talk about the expeditions to Nubia, but it was in the era of the Middle Kingdom of Egypt that they conquered.

In the eighteenth year of the reign was conducted a campaign against Nubia in which the name of Kush is mentioned for the first time. The conquest was at this purely military stage: control of resources and creation of a area controlled by numerous fortifications.

In the Second Intermediate Period of Egypt (between 1780 to 1580 BC) the Egyptians lost control of the area: the Nubians were liberated and destroyed the forts and reconquered the country.

Why did each take control of the other in turn (question 2)

Both towns wanted to conquer the territory that borders the Nile. The river was a source of water and life and an ideal space for the people. Kush was a region located along the valley of the Nile that included the south of Egypt and extended by the north of Sudan. It was a profitable land raw materials, especially in gold, so the Egyptians very early ambitioned to exploit these resources.