Is there some relationship between circuit voltage and the speed of circulating electrons? It is well-known that under an external electric field the work that a charge has to do to go from a point A to another point B is $w=-V_{AB}*q$, this is equal to $\int_A^B{-\nabla V·dl}$ which would equal to the integral of the electric field over the path if the potential vector is constant in time.
However, I have not been able to find any information about the existence of a relationship between the average momentum (or speed) that the circulating electrons have in a circuit and the difference of potential (or electromotive force) existing on that circuit.
As long as there are enough conduction electrons in a material, the same current could be achieved with fewer electrons moving faster or more electrons running slower if the $I=n*e*v$ relationship is kept, where n es the number of electrons per unit of length in the circuit. I am aware that the electrons do not cross the circuit individually but transfer their momentum to the next electron in the circuit, somehow similar to Newton's cradle, but that does not change the fact that we can get the same current if the conduction electrons are more often conducting with less speed or less time conducting with more speed.
It seems to me that if the remaining potential at some point is  $V_{A0}$ , the conducting electrons speed at that point should be at least equal to  $\sqrt{2e/m_e*V_{A0}}$  so that the conducting electrons will be able to retain some speed at the end of the circuit (in the point with less potential).
I wonder if this approach is correct.
 A: 
I have not been able to find any information about the existence of a relationship between the average momentum (or speed) that the circulating electrons have in a circuit and the difference of potential (or electromotive force) existing on that circuit.

There is no such relationship in general. Consider for example a capacitor or a circuit with a capacitor in series. In the dielectric the speed is zero regardless of the voltage.
However, although there is no such relationship in general, there is such a relationship for a conductor obeying Ohm’s law. This speed is known as the drift speed: $$v_d=\frac{I}{neA}=\frac{V}{RneA}$$
So in a conductor if you double the voltage you will double the speed, but the speed itself depends on the free electron density, $n$, of the conductor material.

It seems to me that if the remaining potential at some point is  $V_{A0}$ , the conducting electrons speed at that point should be at least equal to  $\sqrt{2e/m_e*V_{A0}}$

This is not correct. If you consider a simple circuit of a battery and a uniform conductor then the remaining potential decreases linearly from $V$ to 0, but the actual drift speed is constant and depends on $n$.
