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The best AF mode would be to drop that gimbal head and shoot them BIF birds with a hand held camera/lens.
I know that there are people who have been very successful shooting birds-in-flight off of a tripod, but doing so is something I've never been able to do.
With the tripod, the pivot point is about a foot in front of you. On the other hand, when you shoot hand held, you are the pivot point. Thus, when you are shooting hand held, you move your head in a natural way to find and follow the subject in the viewfinder. Finding and following the subject with a tripod and the pivot point out in front of you requires a much less natural movement. Further, when you are shooting hand held, following the action requires only simple rotation of your hips and shoulders. Following the action when shooting off of the tripod requires you to walk around the tripod. Whenever I've tried to shoot birds-in-flight from the tripod, not only do I have a much more difficult time accurately locking in the focus point, but I cannot keep up with the movement of the bird as I am frantically trying to move my body around the tripod, and, if I can't keep the focus point on the subject, I am not going to get in-focus shots regardless of how good the AF system on the camera is or regardless of what setting I'm using.
I commend those who are able to shoot birds-in-flight from a tripod, but they are able to do something that I have not been able to do.