Which particle mediates the Aharonov-Bohm effect? BACKGROUND
The Aharonov-Bohm (AB) effect induces phase shifts between the two paths that an electron could take around an enclosed magnetic field. In radial coordinates, assume that the magnetic field is localized around the origin and that the two paths traced by the electron form two complementary half-circles at radius some R. Assume further that the magnetic field is initially switched off. 
QUESTION
At the moment the magnetic field is switched on, which particle travels outward from the origin towards the electrons' path so as to mediate the phase shift? And at what speed? Clearly, such a particle can't be a disturbance of the electromagnetic field since the magnetic field is restricted to the origin and its vicinity.
 A: Useful physics exercise here would be to study the problem of the solenoid with a current increasing with time. The field is growing inside the solenoid, and so is the field energy there. Where is the energy coming from? Answer is: work done by the current against an electric field. The electric field exists not only at the wire but also throughout space within the region accessible at the speed of light since the current started to change. It's not hard to calculate: you have non-zero $d{\bf B}/dt$ so non-zero curl ${\bf E}$. Integrate this over the interior of a circle of whatever radius you like and convert to line integral:
$$
\oint {\bf E} \cdot d{\bf l} = -\frac{d}{dt} \int {\bf B} \cdot d {\bf S}
$$
The symmetry tells you ${\bf E}$ is in loops around the solenoid, so
$$
E =  -\frac{1}{2\pi r} \frac{d \phi}{d t}
$$
where $\phi$ is the flux in the solenoid. 
So, to answer your question: the influence of the change at the solenoid is carried by this field, and therefore it is mediated by photons (whether real or virtual).
A: It is the electromagnetic field disturbance.  You may have static magnetic field enclosed in the solenoid, but it is not possible for dynamic field. $\mathbf{\dot{B}}=-\boldsymbol{\nabla}\times\mathbf{E}$
So a switching on of the solenoid will produce a spreading wave
AB-effect is purely electromagnetic (and quantum)
A: 
At the moment the magnetic field is switched on, which particle
  travels outward from the origin towards the electrons

You have already answered yourself: magnetic fields are mediated by photons.

And at what speed? 

Photons generally travel at the speed of light :-)

the magnetic field is restricted to the origin and its vicinity

Whoa, this is the issue right here. The range of any magnetic field is infinity. Its strength falls off, but there is no place where it is zero.
I am curious why you believe otherwise, as it seems you are aware that EM is due to photon exchange, and as photons have no half-life, one would naturally assume there is no inherent range limit (unlike, say, the strong force where the mediators have a short life and therefore can't get very far).
