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On a keyboard+mouse setup you have to hold the direction it indicates (using the W,A,S,D keys) and at the same time quickly tap the weak punch button (F key) to drain the red bar that appears. You always have to switch to a new indicated direction half way through a red bar to continue draining it. When the Vulture is killed you want to hold the grip button and tap the jump button until Spencer is ready to fight the next Vulture.
I'm curious about a short cut for speed run purposes, but haven't tried it yet. On the third or fourth Vulture when Spencer is connected to it, but hasn't zipped in to fight it, you can see Joe's special Vulture off to the side. If it is possible to detach from the Vulture and reattach to Joe right then it could save you having to kill the last one or two Vultures before the video with Joe starts. ...or it might not work and you'd die.