The false dichotomy of that stark "were they black or white" is going on in Afrocentrism as well as right here on FR -- Egypt wasn't all one thing.
In their art, they were much more vernacular even while being stylized, typically showing men with darker skin (because they spent more time outdoors) and women with lighter skin (because they spent more time indoors). They also portrayed Asiatics (Semites, mostly) in specific ways, black Africans such that they are recognizable today, Keftiu, Pereset, etc -- and the importance of each human portrayed is often signaled by their relative sizes.
Greeks arrived in great numbers before, during, and after Persian rule, as did other less familiar groups who went where the action was. Roman rule meant former borders were down and transportation was simpler than it had ever been (and perhaps was going to be up until modern times) so there isn't much of a surprise that there were changes in the population's DNA. Roman rule had a nebulous boundary, and Roman influence via trade had a very great reach. A bear that was native to Africa was hugely popular in the Roman "games", and wound up extinct due to that use.
roman mummy coffin portraits
The artistic convention of showing men as darker and women as lighter (because of how much time they spend outdoors) is also found in Bronze Age Minoan/Mycenaean art. There were definitely contacts between the Minoans and the Egyptians—Sir Arthur Evans was able to create his chronology for Minoan civilization based on imported Egyptian artifacts, and Cretans are depicted in 2nd millennium Egyptian art. Also the Thera frescoes show blue monkeys which may have reached there via Egypt (Ethiopia is the northern end of their range).