How do delocalized electrons conduct electricity?

I have learned that for a meterial to be conductive, it must contain free charge carriers, in most cases electrons.

Graphite does conduct electricity parellel to its graphene layers, which is due to the $$p_z$$ orbital of the carbon atoms containing electrons which are not "used" for bonding with other carbon atoms (as it is the case with the elctrons in the $$sp^2$$ hybrid orbitals - these are used to form $$\sigma$$ bonds to other carbon atoms).

The $$p_z$$ orbitals of the carbon atoms interact and form $$\pi$$ bonds whith each other, which results in a conjugated pi system - the electrons are delocalized over the whole layer (one could say that there is one big hybrid orbital formed from the individual $$p_z$$ orbitals.

However, since these electrons are not localized, one cannot define some movement for it - as far as I've understood, the orbital only describes the probability of where you may find the electron if measuring its position. Until then, due to the wavelike behaviour of an electron, it cannot be said to be in one specific place.

So how can delocalized electrons transfer electricity if one cannot really say that they are "moving"? (I asked about graphite, but the same should apply to metals)

Delocalised does not mean stationary. Consider that archetypal delocalised particle the free particle, which we write as:

$$\psi(x,t) = e^{i(\mathbf k\cdot\mathbf x-\omega t)}$$

This is delocalised because the probability of finding the particle is independent of the position $$x$$, however it has a momentum:

$$\mathbf p = \hbar \mathbf k$$

And since it has a non-zero momentum it is definitely not stationary.

In fact we can approximate the $$p_z$$ electrons in your graphene sheet as free particles since they can freely hop from $$p_z$$ orbital to $$p_z$$ orbital across the whole sheet. So the $$p_z$$ electrons in the graphene all have a non-zero momentum just like our free particle. In the absence of an external electric field the average momentum is zero because the electrons are all moving in random directions so there is no net motion. However once we apply an external field we have more electrons with their momenta aligned parallel to the field than in other directions and now we do have a net non-zero momentum i.e. a current.

This applies to metals as well - the argument is exactly the same except that the momentum can point in all directions instead of being confined to a 2D sheet.