I feel it was possibly his wife Zipporah who could have been considered a Cushite.
We notice in Exodus 2:7 it was Miriam who was responsible for Moses to be raised by his mother. Aaron was an older brother born before the command to kill the Hebrew male babies. Since the two were older than Moses they probably thought they should have had more authority than they were given. Remember this was after Aaron made the golden calf. We must be very careful of our thoughts about authority and control. Moses didn't ask for this nor wanted it. He tried to talk God out of it.
When God called the three of them He vindicated Moses as one who He had especially called to do this work.
I wonder why Miriam was given leprosy and not Aaron. I am supposing she may have been the instigator putting it in the mind od Aaron. There is a lesson for us in all of this. Take care of what God has given us to do an no more.