Insofar as board members are compensated by stock grants, they are incentivized to increase the stock price. But if a board member starts trying to do independent verification of what the CEO/team is telling them, that's a sign of mistrust. Because the board primarily interfaces with the executive team, would they know where to start?

That said, I've seen some board members try this out by offering to help/chat with some under-performing teams that they have expertise working with and asking light questions/giving advice... but having been on the receiving end of a conversation like that, it often feels like there is a lot of politics going -- you don't know if someone genuinely wants to be helpful, is vetting your capabilities and fit for the role (based on the trickle-down responsibility), or is trying to do independent fact finding (trickling it back up).

They are not punished if they don't -- this is the right question. Except if they are a major-ish stock holder which is usually not the case (for publicly traded).

In some rare cases the board has retained outside counsel to conduct an independent investigation when there is potential wrongdoing or conflict of interest involving the CEO or other senior executives. The board doesn't receive any extra direct rewards for this, beyond protecting their own holdings and reputations.