Board Thread:Theories and Speculation/@comment-35448581-20160502192848/@comment-25122992-20160506171145

Yeah, Rinne Sharingan has nothing to do with the Byakugan, it's just the result of Kaguya consuming the Chakra Fruit and attaining the power of Shinju, i.e. the chakra itself. I'm not even sure that until getting the chakra, Ootsutsuki eyes had been any doujutsu at all. It's even possible that these were just regular eyes of the Ootsutsuki folk with no special abilities, yet in conjunction with chakra, they were turned into a real doujutsu. Thus, Kaguya gained two doujutsu powers with the Fruit: the power of the Shinju itself (Rinne Sharingan) and the power that awakened in her own eyes thanks to chakra acquirement (Byakugan). These two powers were individually inherited by two children of her, yet while Hamura got his power with no changes (the same Byakugan as Kaguya's), another one (Rinne Sharingan) deteriorated/changed it's form into new doujutsu - Rinnegan of Hagoromo.

Sharingan, the eyes of his older son, is surely a deterioration of Rinnegan's power cause it was technically split apart between the ocular power (Sharingan itself) and the body power (Asura's abilities). Byakugan was separately spreading amongst Hamura's bloodline, and although some power deterioration probably happened, all Hamura's descendants (i.e. Hyuga and Ootsutsuki clans) possess this doujutsu in it's primary form.

So these two eye powers are surely separated from one another. However, there's still the question why Hamura's Byakugan remained unchanged among all his descendants, while Hagoromo's Rinnegan was split into 2 different powers and continued to change it's form between his descendants.