Wheels up (and flaps down) -
After drafting up my answer in Word to paste in, I find robione has beaten me to it! Still, here is my farthing's worth:
In answering this query I am assuming that we are looking at basic aerodynamics and considering a standard training aircraft with inboard trailing edge flaps.
When we talk about a wing drop at the stall, we are talking about a sudden loss of lateral stability.
The severity of the wing drop depends on two factors:
a) The imbalance in lift between the wings
b) The distance of the centres of pressure from the longitudinal axis.
In an unflapped wing, if one wing starts to stall at the tip first, the centre of pressure will start to move inboard. Not only has the lift decreased, but so has the length of the arm to the longitudinal axis. A double whammy!
If, by the use of washout, or root spoilers etc we can ensure that the wing root stalls before the tip, we can ensure that as a wing starts to stall, the initial loss of lift occurs inboard. The centre of pressure will then move out towards the wingtip. This increases the length of the arm and the turning moment derived from the remaining lift.
Right, down with those flaps!
Firstly, you are bang on right that the inboard sections will stall before the outboard sections. This would seem to imply that this would lead to improved lateral stability at the stall. However, there are other factors at work:
a) With flaps extended the centres of pressure move inboard due to proportionately more lift being produced on the inboard section of the wing.
b) At the stall, or even at angles of attack past the stall, the flapped aerofoil can still produce more lift than the unflapped aerofoil, even though the latter is unstalled. What happens is the the coefficient of lift/angle of attack curve moves substantially upwards and slightly to the left. So, even with the inboard, flapped, wing sections stalled, the centres of pressure are still further inboard when compared with the unflapped wing. This leads to decreased lateral stability.
So, what it all comes down to is the lateral instability caused by the inboard movement of the centres of pressure.
Doesn't add a lot to robione's answer really, who put it much more succinctly!