The answer isn’t always clear. The statutes and bylaws usually say one serves as an officer for a term of years and/or until a successor is elected and qualified. Where the officers are elected by the board at the annual meeting of the board (or by the members at the annual meeting of the members), some will change officers in the middle of the meeting at which they are elected. Some will elect the officers with an effective date of the beginning of the fiscal year or some other time after the election meeting. Some will, by tradition, have the new officers assume office immediately after the end of the meeting at which they are elected. If a new officer is appointed to fill a vacancy, normally that person would take office immediately upon appointment. The language of the bylaws, the default provisions of the state nonprofit corporation law, or established tradition should provide a basis for determining the moment the officers first have authority to act.

If board membership depends on being an officer, which would create an ex officio position on the board (i.e. being a director by virtue of being an officer) even though not always called an ex officio position, the term on the board would presumably end when the term of office ends and the new officer has power to act. I don’t recall seeing litigation on the issue and, since most nonprofit boards are not contentious, I assume they work out any problems that may arise. 

I wouldn’t recommend trying to put stuff through a lame duck session of the executive committee after new officers have been selected. It could have long-lasting adverse effects on the functioning of the organization. And most of the changes could be undone as soon as the new officers take over.