I have only finished two of Ron Hess' horn kits, so I am a rank amateur.
That said, I see no reason not to finish each part separately. I think each material needs its own finishing process anyway.
So, I finish each part, then assemble and pin, etc.
Both horns I have done were done with Rit Dye on the horn itself, then later surface treatments. When boiling a horn there would be no way to have the plug on there.
I tried aqua fortis on horn pieces, but get only a pale yellow cast.
That's what I do, but then, my opinion means little compared to real horn makers.
God Bless, Marc