This has gotten me killed recently, and I checked to try myself, and it's scary how fast you can draw, and shoot, to the crosshair without even yet pointing the weapon at the opposing person! (during an interaction)
This is when you shoot as you soon as hold RMB, and the crosshair pops up, allowing you to kill anyone instantly at close range, without them even knowing you had your gun raised in the first place.
The bullet should not follow the crosshair, until the weapon is fully raised, (it should follow gun barrel in process of raising weapon) and once it's fully raised, the bullets can go towards the crosshair.