ANSWER: Some think Moses married a woman of a different race (he was married to a woman with black skin). She was called a Cushite, a term often used in the Bible to refer to African nations. But Cushite also referred to countries south of the land we today know as Israel, some of which were related to the Israelites through Ishmael and the sons of Lot.
"Miriam and Aaron began to talk against Moses because of his Cushite wife, for he had married a Cushite" (Numbers 12:1, NIV)
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 of Israel by obeying God's laws. Based on the Bible, Gentiles who decided to live among the Israelites (during Moses' time and afterwards) could freely worship the Eternal as any natural-born Israelite.
Although any alien or Gentile that was not circumcised could not partake of the yearly Passover ceremony, they still were required by law to do several things. They 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, etc.) 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 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 verses (and others which could be listed), there is NO prohibition against what people define as different "races" from being married.