Cushites is just an anthropological label and has nothing to do with the Kingdom of Kush. If there was a migration brining this language group it would have been around 10,000-6,000 years ago. Well before any of those civilizations.
Wow. Just wow.
https://en.wikipedia.org/wiki/Cushite
"Cushite may refer to :
- Cushitic languages
- the Beja language
- Agaw languages
- Highland East Cushitic languages
- Lowland East Cushitic languages
- South Cushitic languages
- Kerma Culture (present-day southern Egypt and northern Sudan).
- the historical Kingdom of Kush
- a biblical tribal name, see Cush (Bible)
- a Hebrew term for "African", see Cushi"