The following has been said about the first shot in a semi-automatic firearm: "...which I consider to be more representative of its potential group size, since the first shot (hand-cocked) was the one which went wide. It is well known that in any semi-auto weapon, this shot will invariably shoot differently than those fed automatically as the gun fires..."
The person who made this statement originally was commenting on a Franchi semi-automatic shotgun, but research indicates that it also applies to pistols, for both of which such variations in accuracy are not really noticed under normal circumstances, neither are they important. More importantly, though, this effect applies to rifles, where it's impact will be felt during distance shooting.
This phenomenon was originally described in the period around the Second World War, and is related to the difference in headspace, and thus the difference in muzzle velocity. This headspace variance results from the difference in the speed, and the resultant variance in force, with which the action is manipulated in loading.
In order to understand this, one needs to look at the construction of the bolt, and at the locking lugs in more detail. These lugs, with the purpose of locking the bolt in place in the receiver, should allow no backwards movement of the bolt, to prevent it from being blown backward. On the forward side, however, one normally have some play, and this causes the problem, as the bolt can now potentially have additional forward movement. When the bolt or breechlock reaches the point of full locking, it may still go forward until it is stopped by the barrel shoulder or some other stopping point. Assuming achievement of a perfect condition, i.e. no clearance forward of the locking lugs, the forward motion of the bolt would be stopped exactly at the point where the lugs were completely engaged. To obtain this perfect condition, the forward clearance on the bolt would have to be exactly zero. Achieving this on a mass-produced action is almost impossible. Assume therefore a clearance of .005" forward of the locking lugs. Also assume a headspace tolerance on the rifle of .005", in addition to a tolerance of .005" on the length of the ammunition.
Now, returning to the cartridge: When a rimmed cartridge is chambered, there can be no forward movement once the cartridge rim has settled against the chamber face, as seating takes place on the rim. Rimless cartridges however, seat on the tapered shoulder which connects the body of the cartridge to the neck. When the bolt of a bolt-action rifle is closed, however forcibly, movement is basically arrested before it starts to rotate before the cartridge is fully chambered. When a semi-auto is manually cocked, the force on the spring is much less than that imparted by the gases driving it back after the first shot - and the resultant closure of the bolt is thus also with less force. When, however, the second cartridge is chambered, the bolt now slams forward that extra .005" after locking.
Suppose that the cartridge being chambered is on the upper limit of its tolerances ( i.e. longest possible ), and the chamber is on the lower limit of its tolerances ( i.e. shortest possible ). Now the additional forward movement of the bolt will forcibly deform or resize the cartridge shoulder, the results of which may be a working headspace of up to .010" above the theoretical minimum. This will result in higher pressure in the compressed cartridge than that from the manually chambered cartridge.
The rounds chambered by the action of the rifle will be chambered uniformly, but the chambering parameters will differ from that of the manually chambered cartridge. It can be seen that, the closer the tolerances, the less the effect will be. If one can get the perfect condition, with no forward movement of the action after closure, a cartridge chambered by manually manipulating the action will be chambered exactly the same as a cartridge chambered automatically after a shot.
On top of all this movement and deformation, this play may result in the cartridge not being flush with the bolt face, and may even be some distance from the bolt face (This does not apply in the case where the extractor claw keeps hold of the cartridge all the time.). The firing pin blow on the primer will thus lack uniformity, as the distance to the primer will vary, which will then definitely show up in the groups.
It must be remembered that this is not the only reason that group sizes will differ under normal circumstances, but, all else being equal, it should explain that first shot being (slightly) off.