I think the Byakugan does allow certain reactions, based on the victims movements. Only in the Byakugan's case, it's purely psychological, as Neji pwned Hinata, in their discussion. By looking at every little detail in her face, body movements and probably breathing, he was able to determine what she was feeling, what her life is like, and even what kind of a person she is. Well, that could've been Neji's own quirk, though.

And Kakashi worked on Sasuke's body speed, because body speed was apparently required, in order to learn the Chidori.

I think it's very possible that Neji can learn it, too. As well as the Rasengan. Neither of them are blood-limits, which theoretically, make them possible. Provided,the learner has the chakra battery to do so.

Byakugan > Sharingan.

Neji > Sasule

Hyuga > Uchiha