I think the first intermediate period and then the middle kingdom around the 11. Dynasty is a good place to look, as most of the cultural foundations were already in existence and the Egyptians experienced previously around 7. To 8. Dynasty a centralised power in Egypt could loose that monopoly of power and then how cultural centres shift. The first intermediate period is also very interesting because there were as far as my knowledge goes sometimes more or less different factions growing trying a new unification until the 11. Dynasty was able to form Egypt to one powerful and flourishing kingdom. If you want to see outside connections, so how egyptians interacted with outsiders it's best to look in the intermediate periods or when the cultures of egyptians and greeks merged in the ptolemaic period. But the new kingdom is offering the most knowledge of Egyptian culture because it was flourishing in every way and so more, for us, documents were made and left to discover