I would note, too, that there should presumably be a difference between "being able to take off and fly with a load," and "being able to hang onto a load without being plummetted into freefall."
A flying creature/machine is going to need to generate N amoun of power in order to generate lift and take off. This is invariably MUCH higher than the amount of power needed to SUSTAIN flight. So while an encumbered creature couldn't -take off- from the ground, it's not beyond concept that a creature already in flight, taking on a certain amount of encumberance, couldn't continue to fly--or a least, to avoid instantly crashing.
If you have a griffon flying along "fully loaded"--just under Medium encumberance--and, feeling peckish, it suddenly decides to snatch a 5 lb duck out of the air, then going from Light to Medium encumberance isn't going to suddenly cause its wingbones to shatter and the poor bird to plummet to earth helplessly. More likely, is hat it may find itself unable to gain altitude, or slowly descending.
Consider a multi-engine aircraft--say, an old WWII bomber like a B-17. It's flying along on four engines. It can lose one engine and still fly more or less normally--though more slowly and at lower altitude than before. Lose two engines, and it's probably on a one-way ticket down--but it's not as bad off as if it had NO engines. It still has power, and can likely control itself well enough to descend at a slow enough rate to keep its crew alive (assuming it has a convenient landing spot).
Of course, the only time your flying creature/machine is going to have to deal with more load than it can take off with, is in "special" situations like the one above. So this needn't really be a consideration beyond rough on-the-spot type ruling. Depending on how heavy a Warforged is, it's not out of the question that the griffon couldn't carry it down, slowing its descent enough for the warforged (and the griffon) to survive.
Maybe a convenient house-ruling would be, an encumbered beast cannot take off fly, but that a beast with Medium encumberance can make a controlled descent (but cannot increase speed, altitude, or do any maneuver more demanding than a slow turn). A Heavy encumberance rips itself out of the beast's grip, or drags it down into freefall.
