A grapple ends if the target is moved out of the grappler's reach. Bigby's hand loses its grip as soon as the dragon starts to fall, whereupon the dragon is no longer grappled and stops falling.Legendary Resistance doesn't apply to a skill contest. Have your Bigby's Hand grapple the dragon mid-flight.
(If the dragon were being grappled by a physical creature--a roc, for example--I would say that the grappler could choose to fall with the dragon and maintain the grapple. However, Bigby's hand is a spell effect rather than a physical object. There's no reason to believe it's affected by gravity.)
And let's say you do, by some means, cause the dragon to crash. What then? Falling damage is capped at 20d6; an average of 70 damage. That's a punishing hit, but an adult dragon can take it and keep going. Knocking the dragon out of the sky is a major accomplishment, as it should be, but it's not an automatic win.
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