Well, light flicker can be caused by a lot of things really. You can pretty much get rid of most light flicker by shutting out natural lighting, keeping camera settings on manual, wearing dark clothing, sitting in the same spot every frame, etc. However, light flicker can crop up in one situation that you can't really control.

Since Lego pieces are quite reflective, occasionally as you are moving your camera, or even just animating something else in your scene, the bright reflection of a light will become visible to the camera and cause a subtle bloom in your image that tends to have the effect of slightly lightening the whole frame. I don't know if that is what is happening with you, but it is something to look out for. Webcams are more susceptible to this than DSLRs in general also, in my experience. There really isn't anything you can do about it exactly, short of trying to keep light reflections on your bricks out of the camera, which is sometimes impossible.