The Deciding Faktor in whos Passive barrier will be up is dependent on the team initiative. so not the initiavite of the barrier ninja.
For example. lets say team 1 has all refines on 4 and has an onoki. and Team 2 has all refineson 3 but all 5 on the barrier ninja. Team one will get the barrier, cause over all Team one has more initiative and will cast it as 2nd.