L'viyim (Levites) in the bes hamikdash (Temple in Jerusalem) had various roles: some sang, some had overnight guard positions, etc. The kohanim ("priests", if you will) had a rosh bes av (family head) who was officially in charge of them and had a m'mune (appointed one) who assigned positions each day. (If I recall correctly — which I might not — some say that the m'mune and the rosh bes av were the same person. But that's neither here nor there.) Who was in charge of the group of l'viyim that served in the bes hamikdash each week (or whatever their shift was)? Who assigned their jobs?
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The Rambam writes (Hilchot Klei HaMikdash VeHaOvdim Bo 3:12) regarding the division of the Levite families:
He later (7:1) writes that the Memuneh (Temple designee) you mention in the question did not only service the Kohanim. Rather:
He proceeds to describe exactly what each Memuneh did, some working in 'departments' which manage the Kohanim's work and some in 'departments' which manage Levite work.
So essentially the Levites were managed the same way the Kohanim were: by the head of the family of the shift and by the designated staff of the Temple.
It seems that according to Divrei Hayamim, chapter 23 that the leviim were broken up into 24 groups each with their own "Rosh Avot". It is hard to tell if they assigned themselves, or were assigned based on the Kohen in charge of such things. However, it seems strongly implied that it was the Cohen who gave orders to the Leviim who were not singing or on guard duty, or acting as judges and officers.