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. The truth is, Gentiles who lived among the Israelites (in the time of Moses) could worship God as fully and freely as any native citizen.
While no uncircumcised male was permitted to eat the Passover, the strangers (aliens, or Gentiles) living among the natural citizens of Israel were to do several things. They were to keep God's holy Sabbath Day (Exodus 20:10), observe the annual Feast of Tabernacles in the Fall (Deuteronomy 16:13 - 14) and bring the appropriate sacrifice for 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 to obey all the Commandments, Statutes, and Judgments of God's Law (Exodus 12:48 - 49). Other scriptures that state the sojourner, like the home-born, could offer sacrifices and take part in the red heifer ceremony are Leviticus 17:8 - 9; Numbers 15:14 - 16; 19:10.
The stranger sojourning among the Israelites could experience all the blessings afforded by obedience to God's Law. Home-born citizens were forbidden to vex the aliens among them, and were reminded that they had been "strangers" in the land of Egypt. They were therefore to show compassion to the aliens, and were to treat them as they would treat any other citizen. The primary differences between natural citizens and uncircumcised strangers were that the latter could not eat the Passover (unless they were willing to be circumcised) and were not released (if they were bondservants) in the Year of Jubilee (Exodus 12:43, 45; Leviticus 25:45 - 46).
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.