After Yisro brings Tzipora and Moshe's sons back to him at Exodus 18:2-5, they are not mentioned by name again in the Torah. Divrei Hayamim (Chronicles I 23:14 et seq.) lists their male descendants but only tells us that they were normal leviim. Do we know anything more from the Oral Tradition?
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Apparently it didn't go great for them. Moshe's calling took its toll on his wife and kids. In the book of Judges it tells of a fellow looking to open up an idol-worshiping temple, but he needs a priest; he encounters a job seeker who's just a plain Levite --eh, good enough! And this Levite's name is "the son of Gershom, the son of MeNashe" (with a funny-shaped Nun). Our tradition says this funny nun is the verse's way of covering up that in fact it was Moshe. So his grandson wound up as an itinerant-priest-for-hire.
There is mention somewhere that a descendant of Moshe was in charge of finance at the time of David HaMelech. Interestingly, if you read Benjamin of Tudela, in his travel to Bavel (Iraq) in the 12th century, there were 2 Levi dayans (Judges) that has yichus going back to Moshe Rabeinu.