Just a couple quick things to check: I assume you are in AI Servo mode for AF? What AF point selection are you using? Using the automatic AF point selection will enable the focus to track across all AF points, while using zone AF will only track across the points in that zone. So if you are having trouble keeping the bird in the frame, you may want to use the full point spread with automatic AF point selection. There are also a number of AF settings which can be changed in the AF menu (ie sensitivity, tracking rates, point switching, etc). The camera manual is a good starting point to read if you think any of those items need tweaking. Personally, I leave most of those menu AF settings unchanged on my 5d3. You might also check the shot priority in the AF menu (2nd menus page I believe). Make sure you have the focus set as priority rather than the shutter speed (or release timing for the first shot) since generally having proper focus is more important than proper timing.

Stephen