Like the Counsellors above them and the Hands of the Cause before them, Auxiliary Board members hold no administrative authority over any Spiritual Assembly. They cannot instruct a Local Assembly, reverse a decision, or take charge of a community's affairs. Their role is advisory and encouraging.
That constraint is easy to state and harder to inhabit, and it is where the Baha'i design is most tested in ordinary practice. A person appointed from above, visiting a community, carrying the standing of the institution that appointed them and speaking about how the work should go, exercises real influence whatever the formal position says. Baha'i literature addressing the appointed arm returns repeatedly to the manner in which its members are to work, which is itself evidence that the boundary requires maintaining rather than merely declaring.
The structural safeguard is that the two arms answer upward along different lines. An Auxiliary Board member is accountable to the Counsellor who appointed them, not to the Assembly they are advising, and an Assembly is accountable to the body that elected it and to the National Assembly above it. Neither can discipline the other. What holds the arrangement together is consultation, which Baha'i teaching treats as a religious practice in its own right rather than a procedural courtesy.