Answer: Some think Moses married a woman of a different race than he was. He originally had a wife named Zipporah who was the daughter of a Midianite priest (Exodus 2:15 - 21). Your question refers to Numbers 12:1 that states his mate was a Cushite, a term often used to refer to African nations. Bible commentaries disagree whether this verse in Numbers refers to the race or originating people of Moses' first wife or to a second one he had after Zipporah died.
In God’s eyes, it does not matter whether Moses' wife was a Midianite, Cushite or whatever "race" she was. What mattered was whether the "stranger" or non-Israelite was willing to be obedient to God's laws and thus become a citizen of Israel.
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 Israel
Strangers in the land were commanded to faithfully observe God's holy Sabbath Day (Exodus 20:10) and keep the annual Feast of Tabernacles in the fall (Deuteronomy 16:13 - 14). They were also bring the commanded sacrificial offerings when they committed sins of ignorance.
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. Additionally, those living in ancient Israel had to be circumcised if they wished to partake of the Passover (Leviticus 25:45 - 46, Exodus 12:43, 45).
Based on the Bible, there is no prohibition against what people define as different "races" from being married. It therefore does not matter what was the ethnic heritage, lineage, etc. of Moses' wife.