Most of the shooting time is "between the targets" so learning how to use recoil to drive the gun from target to target is key for picking up speed.
Like JT mentioned one of the best ways to do it is to shoot at a cadence. That is, to make your transitions the same as your splits. If the shots in an array are all about the same difficulty, I will always shoot at a cadence. The more you shoot the faster your cadence will become.
Here's a quick example, the first array on this stage in particular.
I believe I read somewhere when doing the transitions, the twisting movement should happen at the knees, and not so much at the waist. Asking the semi-pros, what is your take, use your knees, waist or a little of both?