How can current flow without voltage? The following is an image from my school textbook

How do electrons travel from E to F is voltage (or the J per C of charge) is zero?
I think my issue is a deeper misunderstanding of the nature of voltage. I have learnt that voltage is the potential difference between two points in an electric field. This means it depends entirely on an electrons position in the electric field. So how come 'voltage' (as seen on the graph) decreases when electrons move through a resistor (e.g. light globe and motor), and not when they move between any two points on the wire (say, between A and B)? Both are simply positional changes, yet one results in a loss of electrical potential energy and the other does not.
The only solution I can work out is that the 'voltage' as shown in the graph, i.e. energy per charge, is a combination of electron's kinetic energy and potential energy. As electrons experience a force from the electric field they keep accelerating as the convert PE to KE, but the total amount of energy remains constant (assuming ideal wires). When they encounter a resistor, they lose KE and so total energy decreases. However, with this theory at point E electrons would have neither KE nor PE, and so should not be able to move to point F.
What's wrong in my understanding?
 A: $I = V/R$
For vanishingly small resistance, vanishingly small potential difference can correspond with high current.
In reality there is no such thing as truly zero resistance in an ohmic conductor, so there's no divide-by-zero undefined current, and there wouldn't be truly zero potential difference across $EF$, but it would be too small to measure with a voltmeter or to represent on a graph at a reasonable scale.
For a real-world example: consider a $10 mA$ current across 30cm of fine 20-gauge copper "speaker wire", which has a resistance of about $0.01 \Omega$.
$V = IR = 10^{-4} V$
Which is right around the minimum detection threshold of a typical voltmeter.
A: Your understanding is generally correct, at "0 volts" in the circuit the electrons still do have potentially energy and are moving (getting pushed by the electrons behind them).
Voltage (electric potential) is entirely relative (just like gravitational potential), so a circuit that goes from 8V to 0V is also a circuit that goes from 9V to 1V.
Speaking of potential energy, water flowing is the best analogy for anything circuit related. A river that flows down a mountain into flat ground will keep flowing, getting pushed by the water behind it. If you consider the flat ground to be $h = 0$, then it's potential energy will be $ E = mgh = 0$, but the river keeps flowing.
A: We say that current flows from higher to lower potential, which is true in general*. But it also helps to understand that electrons do not actually travel down the wire the way you expect when you learn this the first time. Electrons are drifting, but at the speed of few mm per second.
This is in contradiction with our intuition on electrical circuits. For example, when you turn the light on, the bulb starts glowing almost instantly. How can this happen if electrons are so slow?
What happens is that electromagnetic (EM) wave actually travels down the wire and carries energy. When there is a load in the circuit, some of the energy is given to the load which results in some voltage drop and the wave keeps on going.

*Can current flow from a lower to a higher potential? Yes, if you use a boost converter - device based on inductor and semiconductors (diodes, transistors) which boosts (increases) voltage from input to output. The current decreases by the same factor, i.e. the power is preserved on input and output.
A: 
I think my issue is a deeper misunderstanding of the nature of voltage. I have learnt that voltage is the potential difference between two points in an electric field... So how come 'voltage' (as seen on the graph) decreases when electrons move through a resistor.

I think the graph is actually plotting the potential at a point and not the potential difference. The contrast between the 2 is that potential is the Potential Energy of a unit positive charge at a single point, while potential difference is the difference in the Potential Energies unit positive charge at 2 points.
Why I am sure? 'Cause if you look at the graph, you can see the $f(x)$ or $y$ coordinate at the point B is 9 Joules per Coulomb. They are saying the energy of the unit positive charge at a point. So $y$-axis has got to be representing the potential, and not the potential difference. It is likely your textbook made a mistake by writing voltage as voltage usually refers to the potential difference.
Now, you will notice that the value of the $y$-coordinate (the Potential at each point) gradually decreases as you go through a resistor or a bulb. That's because the Potential Energy of the unit charge at each point reduces, the reason being that $PE$ is dissipated in the form of Heat and Light energy (mostly heat).
Now I am coming to your titular question. How can current flow without voltage? Easy, the answer is there's no resistance. If there's no resistance, then no one is hindering the flow of electrons. Then no one is there to stop or alter the flow of electrons, then the current remains the same. Kind of like how an object in motion continues to be in that state of motion if there's no friction. Friction is analogous to resistance. So no resistance means no friction to the movement of the electrons and they continue to stay at the same speed.
The potential difference, the voltage, is the work done in carrying a unit positive charge against a resistance. A potential difference is needed in non-ideal conditions. Sort of like how you need another external force to neutralise the friction (key word-sort of. Resistance and voltage, after all, have different dimensions, unlike in the example of friction and external force)
A: In reality, with exception of superconductors, the voltage drop in a conductor carrying current never be zero, that is, there is always some resistance. The mechanical analogy is something sliding at constant velocity on a frictionless surface without requiring work. There is no such thing as a perfectly frictionless surface.
But for the purpose of the  circuit analysis pictured, the resistance of the wires can be considered so small in comparison to the three components that it can be ignored (assumed to be zero).
Hope this helps.
A: In a water hose, the water still flows even though the pressure difference is the constant.
Imagine that charges slowed down naturally (due to internal resistances) while moving. Then you would see charge starting to accumulate somewhere. This would introduce a voltage to even out this accumulation and make the charges move along again. The voltage of zero between two points is thus corresponds to a "balancing" current value - for any higher or lower current, the voltage would increase or decrease to adjust the current back to the "balanced" value.
A: 
So how come 'voltage' (as seen on the graph) decreases when electrons move through a resistor

Because (at steady-state), there is an electric field established inside the resistor

and not when they move between any two points on the wire (say, between A and B)?

Because if the wire has minimal (or zero) resistance, there will be a minimal (or zero) electric field within.
Here's the thing: the field isn't some external object applied to the charges in the wire.  As the charges rearrange in the wires, they create and change the electric field.
If a lot of charges hit a resistor and slow down, then they start building up in large numbers.  The large numbers of charges create an electric field in the resistor that pushes them forward.  If more charges enter a spot than leave, the number of charges in that spot builds up until the field makes the numbers entering and leaving the same.
The final result of all this (for a steady-state DC circuit) is that current at any spot in the circuit is identical throughout.  All charges have the same KE at all times.   In places where they might build up speed (low resistance wires), there is nothing pushing on them.  In places where they might lose speed (in resistors), the field pushes strongly.

'voltage' as shown in the graph, i.e. energy per charge, is a combination of electron's kinetic energy and potential energy.

As the charges KE does not vary (at steady state), the voltage is exactly proportional to the PE of each charge at that point.
