Frame dragging is taking the whole gravity field along and warping spacetime as the object goes through space. Think of a rolling bowling ball on a trampoline stretching the fabric as it rolls. Notice though that the curvature around the bowling ball always stays the same.
I think that would depend on the speed at which the bowling ball is moving across the trampoline. Because the stretching of the fabric takes time. If the speed of the ball is great enough, a complete field around the ball (symmetrical depression) wouldn't have sufficient time to fully establish itself at every position along the path of the ball. You would have non-concentric "stretching waves" bunching up in front of the ball and loosening up behind. Sort of like sound waves from the horn of a fast moving car (i.e. Doppler Effect).