Oswald in the doorway: the blog of the Oswald Innocence Campaign, by Ralph Cinque

So, what is going on with Kennedy? He's having a full-blown adrenergic reaction, meaning that his whole system was in panic, fight or flight, with his adrenal glands pumping out adrenaline.

At this point, he's been shot twice: once in the back at the level of T3 to right of his spine, and once in the throat from the front. Neither bullet exited. 

As Gil Jesus explained, Kennedy was cupping his mouth with his right hand and coughing to dislodge the bullet, and with his left hand, he seems to be pulling on his tie to loosen it. The idea that all that muscular activity was due to reflex, like a knee-jerk, is ridiculous. However, it happened so fast, he hardly had time to think it out consciously. So, I would say that it was really his instinct to survive that was driving it. 

And, by the way, I don't think there is any reason to think that Connally had been hit yet at this point:

I don't want to hear anything about a collar flipping up or any other nonsense. He did not yet having a bullet bore through his chest from back to front. For goodness sake, he would have reacted to that immediately and in a very conspicuous way. He is NOT reacting. Kennedy is reacting. Connally is not.