It sounds to me like a lens AF speed issue too. The 5D3 has fantastic AF capabilities, but if the lens is physically too slow moving to keep up with fast subjects there is nothing the camera can do to help it. My 100-400L is also slow to focus, when compared to my 300 f/2.8L, and can have the same issues. That would explain it locking on at a distance (particularly if you start the AF before the bird is released) then loses it because the lens is not able to adjust quickly enough.
A couple of questions:
Were you taking single shots or in burst mode? If in burst mode, how is your camera set for "AI servo 2nd image priority"?
If that is set to speed (or even equal) then the camera may be shooting too quickly. That should be set to "focus", which should slow down the frame rate to allow the lens more time to adjust between each shot, and the camera should only shoot when focus is achieved. You could try and track the bird, without shooting, until it gets to where you want it, then take a single shot. that should allow the lens time to focus and your one shot should be sharp (assuming a fast enough SS of course). Alternatively, do it the old fashioned way and set the lens to MF, prefocus at a set distance (ideally with a recognisable marker in the scene) and shoot as the bird flies across that point.
Where were you standing relative to the bird?
If you are standing close to the intended landing point, to get a head on shot as it flies, you are going to have some very serious changes in distance to deal with between each shot and the lens will probably struggle with that. If you stand well off to one side and pan with the bird the distance changes much more gradually as the bird is going more left to right (or vice versa) rather than straight towards you, that makes it much easier for the lens to keep up.
I am assuming here that it IS simply missed AF and not softness due to too slow a shutter speed. A fast moving bird coming towards you needs a FAST shutter speed.
As always, posting a couple of images that show the problem makes it much easier for others to see where the problem may lie and give the best advice. If you post one or two it may help narrow down the issue.