Where does the tradition come from that Noah's sons went separate directions?

I want to extend David's answer to your original question and then provide some commentary on perpetuation of the Japhetic et al. migration

Origin

David's answer above is likely the oldest source we have that mentions this tradition. It is also helpful to note that several apocryphal writings account for the same migration patterns, although their respective level of detail varies. For example, the Book of Jasher reads:

And they went and returned every one to his way and to his place, and Noah and his sons dwelt in the land that God had told them, and they served the Lord all their days, and the Lord blessed Noah and his sons on their going out from the ark. (Book of Jasher 6:41)

In the next chapter, these sons are listed in accordance with locale, although the generational gap is much greater than in Antiquities.

In the Book of Abraham (LDS Scripture), this supposition of ancient origin is perpetuated. Note however, that the status of the Book of Abraham is arguably not reputable as a historical document. It's included here because its content is unique.

21 Now this king of Egypt was a descendant from the loins of Ham, and was a partaker of the blood of the Canaanites by birth.

22 From this descent sprang all the Egyptians, and thus the blood of the Canaanites was preserved in the land.

23 The land of Egypt being first discovered by a woman, who was the daughter of Ham, and the daughter of Egyptus, which in the Chaldean signifies Egypt, which signifies that which is forbidden;

24 When this woman discovered the land it was under water, who afterward settled her sons in it; and thus, from Ham, sprang that race which preserved the curse in the land. (Abraham 1:21-24)

Combining these writings, many scholars conclude that common Jewish historical belief around the lifetime of Christ was that the progeny of Noah had dispersed to separate lands and started unique peoples and cultures.

Perpetuation

Although these writings persisted through the Dark Ages, we don't have any clear evidence to suggest they were taught to Christians in Europe. The data we have available suggests that words such as "Semitic" and "Japhetic" (derived from Shem and Japeth respectively) weren't coined until the 18th century. These words were based in linguistics, although they eventually obtained demographical and geographical meaning. Ludwig Schlözer first used the word "Semitic" in a linguistic text in Repertorium, vol. VIII (Leipzig, 1781), p. 161. and the word came into common use with Eichhorn a few years later.

These terms were very extremely pervasive, surviving up to the modern day. One reason for this pervasiveness was the exceptional skill of the linguists who coined the terms, because their theories ended up being mostly right. For example, most of the Japhetic languages really are related to one another although it's difficult for most people to see the similarities between Portuguese and German.

Modern-day academics has tried to replace some of the Biblical terms as available data grows and expands to include languages not previously included. Semitic languages are now much more narrowly restricted than they were previously and Japhetic is now an antiquated term which has been replaced by "Indo-European"

Noah's son did not disperse after the flood until, God mixed up their languages because of the tower of Babel, but it is probable that they were all still alive, since the tower of Babel was so soon after the flood.

You may be interested in a lesson I taught to my adult Sunday school class, which I have appended below:

From the flood to the tower of Babel

Genesis 5:29 and he named him Noah, saying, "This same will comfort us in our work and in the toil of our hands, because of the ground which Yahweh has cursed."

Genesis 5:30 Lamech lived after he became the father of Noah five hundred ninety-five years, and became the father of sons and daughters.

Genesis 5:31 All the days of Lamech were seven hundred seventy-seven years, then he died.

Genesis 5:32 Noah was five hundred years old, and Noah became the father of Shem, Ham, and Japheth.

As stated it could be reasoned that;

these were triplets since it says that Noah was five hundred years old when he became the father of the three.

However if they were triplets then it becomes hard to explain Genesis 11:10 which says that Shem was 100 years old 2 years after the flood.

Therefore it is necessary to study a verse in relation to other verses in the Bible, and they may relate to other chapters or even other books. The value of a good concordance is shown in this situation, by looking up Shem in the concordance we learn that Shem is also in Genesis 11:10.

The flood

Genesis 7:11 In the six hundredth year of Noah's life, in the second month, on the seventeenth day of the month, on the same day all the fountains of the great deep were burst open, and the sky's windows were opened.

Genesis 7:24 The waters prevailed on the earth one hundred fifty days.

Genesis 8:13 It happened in the six hundred first year, in the first month, the first day of the month, the waters were dried up from off the earth. Noah removed the covering of the ark, and looked. He saw that the surface of the ground was dried.

Genesis 8:14 In the second month, on the twenty-seventh day of the month, the earth was dry.

Since Noah is now 601 years old, at least one of his sons must now be 100 years old. Most Bible scholars think Shem was the oldest because he is named first, however; since we know that Shem was 100 2years after the Flood, and that Noah was 500 when he became a father; and that he was 601 at the end of the Flood; either Ham or Japeth was probably the elder brother.

After the flood

Genesis 10:1 Now this is the history of the generations of the sons of Noah and of Shem, Ham, and Japheth. Sons were born to them after the flood.

Genesis 10:2 The sons of Japheth: Gomer, Magog, Madai, Javan, Tubal, Meshech, and Tiras.

Genesis 10:3 The sons of Gomer: Ashkenaz, Riphath, and Togarmah. Genesis 10:4 The sons of Javan: Elishah, Tarshish, Kittim, and Dodanim. Genesis 10:5 Of these were the isles of the nations divided in their lands, everyone after his language, after their families, in their nations.

As of this the third generation after the flood There are different Languages so the Tower of Babel must have been built by this time.

Genesis 10:6 The sons of Ham: Cush, Mizraim, Put, and Canaan.

Genesis 10:7 The sons of Cush: Seba, Havilah, Sabtah, Raamah, and Sabteca. The sons of Raamah: Sheba and Dedan.

Genesis 10:8 Cush became the father of Nimrod: he began to be a mighty one in the earth.

Genesis 10:9 He was a mighty hunter before Yahweh. Therefore it is said, "Like Nimrod, a mighty hunter before Yahweh."

Genesis 10:10 The beginning of his kingdom was Babel, Erech, Accad, and Calneh, in the land of Shinar.

Again Nimrod is the third generation after the flood. His Kingdom included Babel

Genesis 11:10 This is the history of the generations of Shem. Shem was one hundred years old, and became the father of Arpachshad two years after the flood.

Genesis 11:11 Shem lived after he became the father of Arpachshad five hundred years, and became the father of sons and daughters.

Genesis 11:12 Arpachshad lived thirty-five years, and became the father of Shelah.

Genesis 11:13 Arpachshad lived after he became the father of Shelah four hundred three years, and became the father of sons and daughters.

Genesis 11:14 Shelah lived thirty years, and became the father of Eber:

Genesis 11:15 and Shelah lived after he became the father of Eber four hundred three years, and became the father of sons and daughters.

Genesis 11:16 Eber lived thirty-four years, and became the father of Peleg.

Genesis 11:17 Eber lived after he became the father of Peleg four hundred thirty years, and became the father of sons and daughters.

Genesis 11:18 Peleg lived thirty years, and became the father of Reu.

Genesis 11:19 Peleg lived after he became the father of Reu two hundred nine years, and became the father of sons and daughters.

From the flood to the end of the life of Peleg was only 340 years

two years after the flood..........2.................2

thirty-five years.................35................35

thirty years......................30................30

thirty-four years.................34................34

thirty years........................................30

To Tower of Babel................101

to end of Peleg's life...........340

Genesis 10:25 To Eber were born two sons. The name of the one was Peleg, for in his days was the earth divided. His brother's name was Joktan.

For in his days the earth was divided. That is between the descendants of Shem, Ham, and Japheth this was accomplished by God mixing up their languages at Babel. Since Shem lived another 500 years after he became the father of Arpachshad, He was still alive at the time of the Tower of Babel and even for part of the life of Abraham. As for Ham and Japheth we are not told how long they lived after the flood, but since it took place probably within the span of from about 100 to 200 years after the flood it is likely they were still alive.

Genesis 9:28 Noah lived three hundred fifty years after the flood.

Genesis 9:29 All the days of Noah were nine hundred fifty years, then he died.

note also that Noah lived after the flood 350 years, to age 950, so he was still alive at the Tower of Babel, and that he outlived Peleg by 10 years.

What else could be learned while studying?

Excerpt from:

Antiquities of the Jews by Flavius Josephus

Now the sons of Noah were three, - Shem, Japhet, and Ham, born one hundred years before the Deluge. These first of all descended from the mountains into the plains, and fixed their habitation there; and persuaded others who were greatly afraid of the lower grounds on account of the flood, and so were very loath to come down from the higher places, to venture to follow their examples. Now the plain in which they first dwelt was called Shinar. God also commanded them to send colonies abroad, for the thorough peopling of the earth, that they might not raise seditions among themselves, but might cultivate a great part of the earth, and enjoy its fruits after a plentiful manner. But they were so ill instructed that they did not obey God; for which reason they fell into calamities, and were made sensible, by experience, of what sin they had been guilty: for when they flourished with a numerous youth, God admonished them again to send out colonies; but they, imagining the prosperity they enjoyed was not derived from the favor of God, but supposing that their own power was the proper cause of the plentiful condition they were in, did not obey him. Nay, they added to this their disobedience to the Divine will, the suspicion that they were therefore ordered to send out separate colonies, that, being divided asunder, they might the more easily be Oppressed.

Now it was Nimrod who excited them to such an affront and contempt of God. He was the grandson of Ham, the son of Noah, a bold man, and of great strength of hand. He persuaded them not to ascribe it to God, as if it was through his means they were happy, but to believe that it was their own courage which procured that happiness. He also gradually changed the government into tyranny, seeing no other way of turning men from the fear of God, but to bring them into a constant dependence on his power. He also said he would be revenged on God, if he should have a mind to drown the world again; for that he would build a tower too high for the waters to be able to reach! and that he would avenge himself on God for destroying their forefathers ! Now the multitude were very ready to follow the determination of Nimrod, and to esteem it a piece of cowardice to submit to God; and they built a tower, neither sparing any pains, nor being in any degree negligent about the work: and, by reason of the multitude of hands employed in it, it grew very high, sooner than any one could expect; but the thickness of it was so great, and it was so strongly built, that thereby its great height seemed, upon the view, to be less than it really was. It was built of burnt brick, cemented together with mortar, made of bitumen, that it might not be liable to admit water. When God saw that they acted so madly, he did not resolve to destroy them utterly, since they were not grown wiser by the destruction of the former sinners; but he caused a tumult among them, by producing in them divers languages, and causing that, through the multitude of those languages, they should not be able to understand one another. The place wherein they built the tower is now called Babylon, because of the confusion of that language which they readily understood before; for the Hebrews mean by the word Babel, confusion.

Life after the flood

0..............100...........200.............300...............400................500

Flood

Noah...................................................350

Shem................................................................................502

+2 yrs Arphaxad..........................................................440

+37 yrs Shelah.................................302

+67 yrs Eber....................................................................470

+101 yrs Peleg..........................................340