They could end up acting like most mob's avatars do (except in Dynamis) where when they use Astral Flow their summon appears just to use their special ability and disappear. Based on the wording (which isn't very reliable anyway) of the notes, the part about how Astral Flow will end upon the Avatar's release, I'm thinking the situation where the avatar appears only momentarily to do the move.
This would be akin to giving us, in essence, two more 2hours. If you take it like invincible, benediction, or those other 'instant' 2hours, it appears to make some sense then. Think of Alexander being a party wide Invincible or perhaps Odin as being like a Majin Gakure (of course this is just an analogy).
The only case in which it might not be this is, if the avatar releases itself or we still release it. If we don't, then we can go on to say that perhaps there is something else they do, otherwise it can be assumed that they have one move, it is instant, and Astral Flow ends immediately upon their disappearance.