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Translations: Gintama 507 (2)
Exactly how is it an assumption? The Kyuubi being sealed within him doesn't change anything, otherwise Naruto would have already had the Kyuubi's chakra when he was born. Not to mention that as far as we saw, Kirabi's entire family held the Hachibi, yet had no easy time using it. And Gaara was the only one of his siblings that could handle the Ichibi, despite them being of the same family. Point is, that Naruto himself is proof that the chakra not only doesn't get passed down, but it doesn't make one "more compatible", else Minato wouldn't have had to split the Kyuubi's chakra.
And the child wouldn't be born with Sage Mode. That would require being taught in this case, and by no means a sure thing. And technically, Sasuke's child would have a much better chance of being able to gain Sage Mode, since Sasuke possesses Juugo's genetics which make gathering natural energy much easier.
Kyuubi is sealed inside of Naruto. Naruto is just able to use the chakra as he wants because Kurama allows him (because of Madara etc.).
We haven´t seen a byakugan develop in any kind.
So a person with both of Uzumaki and Hyuuga genes could see very good (far+360°+tenketsu) and would a strong chakra + sealing jutsu.
What the Person uses the Chakra for can be diffrent (healing like Karin -> wider form like Tsunade, Mito also had the seal on the forhead) or fast and precise Taijutsu.
Maybe some strong Ninjutsu.
A person with Uzumaki and Uchiha genes would have a strong chakra and be able see very fast movements (good against taijutsu), to copy movements, see chakra, able to see genjutsu and use it better, can read plate in Naka no Jinja (maybe way to Rinnegan). MS user can use diffrent strong some of the strong techniques (Amaterasu, Tsukuyomi, Susanoo, Kamui, etc. etc.). BUT you can loose your vision by overusing it (maybe not if you have the strng chakra from Uzumaki?Dunno.)
Strong chakra + sealing Jutsu.
So they could use anything i think...
Whos stronger goes in my opinion to the questions who of Uchiha and Hyuuga are stronger, cause we can see how diffrent Uzumaki chakra can be used (Naruto, Karin, Kushina).
And somehow, naruto's body was able to adapt in absorbing the natural energy. If he keeps on doing that, naruto's family will definitely become like the jugo's clan. His son/s will automatically able to absorb the natural energy.
What adaption? Learning Sage Mode like normal isn't adaption. If that was the case, then Naruto would be capable of automatically going into Sage Mode. Juugo's clan dwell in a area where the natural energy was strong for generations. Naruto only spent time in such an area for a couple of days. He would literally have to go live with the toads, and even then it wouldn't be his son but future descendant that benefits.