What happened to nations descended from Noah’s son Shem?
The 70 Biblical nations of the world came from and are named after the 70 descendants of the three sons of Noah which are listed in Genesis 10:1-32. Of these, 26 were descended from Shem, Noah’s youngest son. 
These are named in Gen. 10:21-31. For a complete commentary on the identity of these nations in modern terms, see Artscroll’s Bereishis (Genesis), vol. I, pp. 308-332. Many of the points below are taken from this source.
The Asian continent was originally apportioned by Noah to Shem and his descendants. This included the land that is now Israel, but they were later forcibly displaced from there by the Canaanite nations, who descended from Ham, Noah’s middle son (Gen. 10:6). (See Rashi’s comment on Gen. 12:6.) Shem was the original king of Jerusalem (which was originally called “Salem”), and he became known as Malkizedek (“Righteous King,” Gen. 14:18).
We will now enumerate the 26 Biblical nations descended from Shem, which are included in the total of the 70 nations of the Children of Noah.
Shem had five sons who became nations
(1) Elam: Their territory was between Shushan (in Persia) and Media. In the days of Abraham (before Abraham had children), the nation of Elam had a king named Chedarlaomer (Gen. 14:1). He ruled by force over the five kingdoms of the metropolis of Sodom and Gomorrah. When those servant kingdoms rebelled against him, Chedarloamer allied himself with three other kings (including Amraphel = Nimrod) to fight and defeat them, as described in Gen. 14:1-16. Abraham subsequently fought and defeated the four kings in order to save his nephew Lot, whom they had kidnapped from Sodom.
(2) Asshur: Asshur became the nation of Assyria, located to the north of Babylon. An individual by this name is mentioned in Gen. 10:11-12, in the story of the Tower of Babel. Rashi comments that when he saw that his children were listening to the wicked Nimrod and rebelling against G-d by building the Tower, he departed from them. He built several cities, including the famous Nineveh, which was known as “a great city unto G-d” (i.e. dedicated to G-d) (Jonah 3:3).
(3) Arpachshad: According to Josephus, he was the ancestor of the Chaldeans. They first lived in the coastal area of the lower Euphrates, and later moved inland. Ur, the birthplace of Abraham, was one of their main cities.
(4) Lud: Josephus seems to identify them with the Lydians of Asia Minor.
(5) Aram: These are the Arameans, who lived in the area of Syria, with Damascus as their capital.
Aram had four sons who became nations
These are: (6) Uz, (7) Hul, (8) Gether and (9) Mash.
Hul: Joshephus identifies this land as Armenia.
Mash: There is a Mount Mash in Mesopotamia that may have been named after this nation.
From Arpachshad to the sons of Eber
Arpachshad had one son who became a nation – (10) Shelah, who was the father of Eber (or Ever).
(11) Eber: Eber was the great-grandson of Shem. He, along with Shem, received prophecy and was one of the great righteous people of the generations. Eber and Shem established an academy of Torah learning. They both outlived Abraham. Eber became a nation, and he had two sons who became nations – (12) Peleg and (13) Joktan.
Peleg: Abraham was descended by 5 generations from Peleg. The dispersion of the nations from the Tower of Babel in Babylonia (present-day Iraq), occurred in the year of Peleg’s death. (See Rashi’s comment on Gen. 10:25.) There is a city of Palga at the junction of the Euphrates and Chaboras rivers.
Joktan: He had the following 13 sons. Most of them are associated with nations that settled in the area of Arabia. Genesis 10:30 states, “their settlement was from Mesha, as you come to Sephar, the mountain of the east.” In “Antiquities of the Jews,” Josephus wrote, “These inhabited from Cophen, an Indian river [now called the Kabul River], and in part of Aria adjoining to it,” which is in eastern Afghanistan.
The sons Joktan who became nations
(14) Almodad
(15) Sheleph
(16) Hazarmaveth: He is identified with Hadarmaveth in Southern Arabia.
(17) Jerah
(18) Hadoram: Doram is the name of a fortress in Southern Arabia.
(19) Uzal: This is the original name of the San’a, the Arabian name for the capital of Yemen.
(20) Diklah: Diklah is used to refer to the palm-tree region in Judea.
(21) Obal: The Samaritans refer Obal to the Mt. Eibal region in Samaria, near Shechem.
(22) Abimael
(23) Sheba: Sheba is mentioned as a distant, wealthy people who were renowned for their gold, precious stones and frankincense. The Queen of Sheba became famous through her meeting with King Solomon. (I Kings 10:2,10) Some identify them with the Sabaeans of Southwest Arabia.
(24) Ophir
(25) Havilah: Havilah is to the southeast of Arabia toward the Persion Gulf and India. There is a place Havilah in Bahrein on the Persian Gulf.
(26) Jobab: Jobab is identified as the Egyptian coastal city of Jobabiti.
After the Tower of Babel
This completes the 26 nations descended from Shem before the Tower of Babel that are listed in Genesis 10:22-31. But were there some of the original 70 nations, perhaps including some descended from Shem, that never succeeded in establishing their own geographic areas? The answer is yes:
G-d caused the original nations to disperse by dividing them with different languages because they rebelled against Him with the Tower of Babel (Genesis 11:1-8). G-d simultaneously sent physical destruction against the people and their tower: He sent a massive storm of hail onto the people (Tur on Genesis 11:8), and a tidal wave from the sea that drowned almost half of the population (Midrash Genesis Rabbah 38:10).
The Midrash (ibid.) says that in the destruction, 30 of the 70 national families were wiped out, leaving only 40, and the 30 that were destroyed were eventually replaced by the 30 nations that arose later from the descendants of Abraham. Of these, 16 were from the 16 children of Abraham and Keturah (Genesis 25:2-4); Abraham sent those children away to the east. Twelve were from the 12 sons of Abraham’s son Ishmael; they settled between Egypt and Assyria (Genesis 25:13-18). So that accounts for 28 nations.
Although the opinion in Genesis Rabbah ibid. cites Genesis 25:23 to identify the remaining two as the nations that came from Abraham’s son Isaac – the Israelites from Jacob and the Edomites from Esau – that was not a permanent situation. The Israelites who survived their period of exile in Egypt, who then became the Jewish people at Mount Sinai, were separated by G-d from the Gentile nations. For example, during the Temple services for the 7 days of Sukkot, 70 bulls were offered to bring blessings to the 70 Gentile nations, and on the next day, Shemini Atzeret, the one bull that was offered represented the Jewish people.
Furthermore, the nation of Midian that descended from Abraham and Keturah was destroyed during the last (40th) year that the Israelites remained in the wilderness after the Exodus. So it may be appropriate to say that the two Gentile nations that completed the number of 70 – before the Israelites entered their promised land – were those descended from Lot, the son Abraham’s brother Haran. Lot, through his two daughters, fathered Moab and Ammon after the destruction of Sodom and Gomorrah (Genesis 29-38).
Additional notes
The Biblical nations of Moab and Ammon, on the eastern border of the Land of Israel, are the Kenizzite and Kadmonite who are prophetically listed in Genesis 15:19 (Edom is the Kenite, the third nation that is listed there). In other words, at the time that G-d spoke to Abraham in Genesis 15:19, the forefathers of the Kenite, Kenizzite and Kadmonite nations, whose lands would lie between the Jordan and Euphrates rivers, had not yet been born!
Abraham’s brother Nahor was the grandfather of Laban and Rebecca (the wife of Isaac). Laban was the father of Rachel and Leah (from his main wife), and Bilhah and Zilpah (from his concubine); they became the four wives of Jacob. The wicked prophet Bilaam (Numbers chs. 22-24) was descended from Laban.
Abraham was personally acquainted with Noah and was taught by him, and he was 58 years old when Noah died at the age of 950. (Note that the numerical value of Noah’s name in Hebrew, No’ach, is 58, hinting that Abraham was the one who took over Noah’s holy work.) Noah was the third in a line of transmitted Oral Tradition of the kabbalistic secrets of creation, and Abraham was the fourth; the line of personal transmission was from Adam to Lamech (Genesis 5:25), to Lamech’s son Noah, to Abraham. From Abraham, those teachings were transmitted to Isaac, then to Jacob, and then to Jacob’s 12 sons. Isaac and Jacob also spent time learning in the yeshiva (academy) of the prophets Shem and Eber.
