Jere Lipps and I suggest that display was involved in the evolution of flapping flight as well as in the evolution of feathers. Display provided long, strong feathers on arms and tail. Successful display was increased by lengthening the arms, especially the hand, and by actively waving them, perhaps flapping them vigorously. Flapping in display would have encouraged the evolution of powerful pectoral muscles. But a threat display must not be seen as an empty bluff.
Fighting is the last resort.
Living birds often fight on the ground, even those that fly well. Beaks and feet can be used as weapons too and are most effective when used in a downward or forward strike. A strong wing flap, directed forward and downward, is also the power stroke that gives lift to a bird in takeoff. Lipps and I suggest that strong wing flapping is a simple extension of display flapping, encouraged by fighting behavior. Powerful flapping used to deliver forearm smashes could have lifted the bird off the ground, allowing it also to rake its opponent from above with its hind claws.