Moshe made everyone drink from the ground-up golden calf waters (Sh'mos 32:20). Did he force the Levi'im as well?
In the beginning the Avodah (service) of Karbonos was with the bechor (first-born -- see the mishnah in the last perek of Zevachim. The avodah was done by the bechorim until the Mishkan was built) ...
What is the source, earliest recorded record, and reason many who are Kohanim and Leviyim write after their name HaKohen or HaLevi?