The Ancient Egyptians were, by our standards today, a "Mixed" people. We might refer to them as "Mediterranean", "Middle Eastern", "Brown", or any number of things to get over the fact that they seem to blur the line between the standard Black-White dichotomy. The point is, I've seen North African "caucasoids" that could pass for Ancient Egyptians, I've seen black Africans (typically East Africans, like Ethiopians and Eritreans) that could pass for Ancient Egyptians, North and South Indians that could pass for Ancient Egyptians, and even Latinos that could pass for Ancient Egyptians. You cannot call them "white" or "black" - two distinctions that emerged relatively recently. To the Egyptians, what determined "race" wan't phenotype, but culture; if you were raised in Egypt, spoke Egyptian, and acted like an Egyptian, then that's what you were. Period.
To answer your question, the 25th Dynasty kings regarded themselves as the guardians of traditional Egyptian religion and kingship, and the Egyptians were overwhelmingly accepting of them because they had brought order to the country after a sustained period of civil strife. However, though the Kushite kings adopted all the trappings of Egyptian monarchy (not a huge step, mind you, as Kushite culture was highly Egyptianized as a legacy of Egyptian imperialism by this point), they still kept their capitals at Napata and Meroe, beyond the borders of Egypt. If the 25th Dynasty can fight off the Assyrians, and prevent them from sacking Waset (Thebes), then they will not be inclined to withdraw to their homeland and will continue ruling, I suspect, much like any other stable Egyptian dynasty. They may even start trying to expand their empire into the Levant, taking the fight to the Assyrians, so to speak.
All dynasties in Egyptian history are ephemeral, however, and soon they may have to deal with Babylonians, perhaps alt-Persians (maybe Medes?) and of course the Egyptians won't appreciate being ruled from Nubia (unless the Kushites move their capital into Egypt), and will probably eventually rebel.