IMO it is about a process. This is how I was taught. I am sure there are variations in technique. But I think this is the jist of it.
The short version is looking, indexing, driving the weapon to the firing position from a low ready chest level retention or a holster to weapon eyes level, driving the weapon forward picking up the sights as you drive the weapon forward to the point shoulder firing position and finger on trigger when sights on and decision to fire is made.
The long version:
You naturally look at something and use your peripheral vision. If you were to point at something you don't swing your arm to point without first looking at the thing you are going to point at. You look, index and point. It is what you do naturally, so when shooting you should use that as a basis, since you are already good at it.
When doing target transitions and when acquiring the first pop up target, whether moving or standing still, you look with your head / eyes focus on a small spot, then drive your weapon sites to that spot. It is important to go eyes on target then weapon on target in that order.
Nobody walks around with the weapon at arms length or pointed down at the ground ( or up ) arms extended like they do in 1970 cop shows.
The weapon is held in a low ready retention position, upper chest level, your head moves with your eyes and you use your peripheral vision for movement. When a target is noticed you look at the target with your eyes, index your head, ID your target while moving you body index to the target and bringing your weapon up from the low ready retention to the firing position ( driving the gun sights on if using the sights) which may be 1 arm bent elbow contact firing and brace / defensive position, a 2 hand or 1 hand bent elbow retention firing position, or a 1 or 2 hand point shoulder firing position, sights on spot on target, finger on trigger when on target and decision to fire is made, fire.
Just like the spinning ice skater, you turn slower with arms extended compared to arms in close. Indexing your body to the threat with the weapon at a low ready retention position also keep the weapon in a retention position until you decide if you need to shoot from a retention position or from a more accurate distance dependent point shoulder position using the sights.Either way, the weapon is moved from the low ready retention position and 'driven' to the firing position for a pistol, one arm bent elbow retention firing position, 2 hand bent elbow firing position or a 1 or 2 hand point shoulder firing position by bringing the weapon up to eye level with bent elbows then driving the weapon forward to the point shoulder firing position picking up the sights and placing them on the spot on the threat in a deliberate aggressive manner being able to shoot sights as required even before your arms are straight out.
These movements including order, need to be muscle memory and done re-actively.
Using a rifle is similiar.
Speed is no wasted movements.
This may or may not align with how others describe it or explain it or teach it., this is what I have been taught for a long time. maybe there is something new out there, maybe not.
I might even be wrong. YMMV