How do I pass down a Moon Ball from one Eevee to another, and will it still work with a female Eevee in Moon Ball and a Ditto in a Poke Ball?
When you breed a male and female of the same species (read: same pokedex number) you can pass down their pokeball with a 50:50 chance. If they're different species then it's the female's ball that gets passed down (or the male's when breeding with a Ditto). The only exceptions are the masterball and cherishball, which can't be passed down. Offspring will come out in a standard pokeball.
Source
The Eevee would have to be female in order to pass it down if you're breeding with another species, if you're breeding with another Eevee in a Moon Ball it wouldn't matter, and male to pass it down if you're breeding with Ditto.
Male + Female of Same species = 50/50 chance for either Pokeball (Offspring is the same species as the parents of course)
Male + Female of Different Species = Female Pokeball (Offspring is the same species as the Mother)
Anything + Ditto = non Ditto Parent Pokeball (Including Genderless Pokemon, Offspring is the same species as non Ditto Parent)