ANSWER: Some think Moses married a woman of a different race than he was. He originally married a woman named Zipporah (see our article on his wife) who was the daughter of a Midianite priest (Exodus 2:15 - 21). Your question refers to Numbers 12:1 which states his wife was a Cushite. This is a term often used in the Bible to refer to African nations. Bible commentaries disagree whether Numbers refers to the race or originating people of Moses' first wife or a second one he had after Zipporah died.
In many ways it does not matter whether Moses' wife was a Midianite, Cushite or whatever "race" she was. The prohibitions placed on the children of Israel regarding marriage to people other than Israelites had to do with the religious practices of those people and whether the "outsider" was willing to become part of the nation by obeying God's laws.
Although any alien or Gentile (a term used for anyone of a race different that Israel) that was not circumcised could not partake of the yearly Passover ceremony, they still were required by law to do several things.
Strangers in the land were commanded to faithfully observe God's holy Sabbath Day (Exodus 20:10), keep the annual Feast of Tabernacles in the Fall (Deuteronomy 16:13 - 14) and bring the commanded sacrificial offerings when they commited sins of ignorance (Numbers 15:24, 26, 29 - 30).
Note that God makes no distinction between those who are Israelites and those who are "strangers" (Gentiles, aliens, sojourners, those who were not of Israel by "race") in regard to the need to offer sacrifices, the ability to receive forgiveness of sins, and even the punishment for presumptuous sins!
Strangers in the land were also commanded to observe and fast on the Day of Atonement (Leviticus 16:29) and additionally adhere to all God's Commandments (Exodus 12:48 - 49).
Other scriptures that state the sojourner, like the person born in the land, could be an active part of the sacrificial services (like the red heifer ceremony) are Numbers 15:14 - 16, 19:10 and Leviticus 17:8 - 9.
Those "strangers in a strange land," the non-Israelites of another race living in the promised land, received the same blessings as those natural citizens who obeyed the Eternal.
Citizens born into the land were forbidden by God's law to "vex" (oppress or otherwise mistreat) those aliens around them as they, too, were at one time strangers in Egypt (Exodus 22:21, Leviticus 19:33). Mercy and compassion were to be shown to aliens.
The only "real" difference between those living in ancient Israel was that a person had to circumcized in order to eat Passover and that aliens were not automatically freed, if they were slaves, during the Jubilee year (Leviticus 25:45 - 46, Exodus 12:43, 45).
Therefore, based on the Bible, there is no prohibition against what people define as different "races" from being married. It does not matter who Moses' wife was or her "race" even if it could be 100% proven.