Best interests test
The number one duty of any board member is to act in the interests of the organisation he/she is overseeing. This means that you must not allow your personal interests, or those of anyone else, to override the interests of your organisation - even if you've joined the board as the representative of another group, or another organisation, or another board.
Some constitutions, for example, make a place on the board for a representative elected by the staff. Such a staff member can inform the board of the feelings of the staff about various matters, or can tell the board the arguments that the staff see as persuasive. It can certainly be argued that such an arrangement is a valuable way of getting into board debates an important perspective that might otherwise not be properly heard.
At least in theory, however, the staff representative cannot be bound to vote the way the staff want them to. When they put on the board hat they have to take off the staff delegate hat. They are obliged to make their own decision independently, ignoring any instructions and disregarding any other interests.
Similar rules apply to other aspects of board practice. A committee member is not permitted to breach committee confidentiality, where the board has decided to enforce it, to report to their constituents, and may not claim that they have other responsibilities that override confidentiality.
In practice it is almost impossible to establish that a person is acting under instruction and thus acting improperly. If they say "I exercised my independent judgment and coincidentally came to exactly the same conclusion as the staff association" there is no way to disprove them. For this reason, these issues seldom come to court, and arguments about them in the board are rarely fruitful.