Why do electrons move as if on conveyor belt in a circuit?
For an excellent in-depth discussion of this part of circuit-physics, see "Matter and Interactions" textbook materials, Chabay and Sherwood, in particular this paper about electrons and circuits (pdf)



How do the electrons know?  The communicate with each other via e-fields; via changes in voltage.  From an engineering viewpoint, it's all about voltage-dividers, plus tiny capacitors.

Instead of building a circuit with wires, use 1-ohm resistors for your conductors, and connect 1pF capacitors between every part of your circuit.  These capacitors represent the voltage-fields found in the space around the conductors.

Initially all the capacitors have zero volts across them.  Now, when the power supply is suddenly connected, how fast do the capacitor voltages shift?  They all adjust themselves with RC less than nanoseconds.  A 'voltage-wave' propagates across the circuit at essentially the speed of light.  (This actually happens.  All conductors, electrons, and components are communicating with each other, essentially instantly: nanoseconds.)

After the capacitor-charges have stopped changing, all of your 1-ohm conductors will have a (small) voltage along them, therefore a lengthwise e-field exists inside.  This field applies a force to the copper's electrons, and they all move in response, creating a particular current.  (In a single wire, all the electrons move as one, like pulling on a chain, or like pushing a rod.  Wires contain invisible columns of electrons.)  The current in each resistor-wire is proportional to V/R: the voltage along each "wire," divided by the small wire-resistance.

The voltages throughout your network of 1-ohm resistors are self-adjusting, just like any voltage-divider.   These voltages then determine the currents.  The voltages tell the electrons how to flow.

Here's the key question: what happens when we remove one long wire (remove it, or even short out its resistor?)  All the capacitor-voltages will readjust themselves instantly.  A "wave of readjustment" propagates across the entire circuit.  The node-voltages are all different now.  This informs all the electrons inside the 1ohm resistors to change their speed (their current.)

Key point:  as with plumbing pipes, the electric current is proportional to the speed of the particles.  High current is fast electrons.   Zero current is just halted electrons.  And AC is like an oscillating drive-belt, where the charges wiggle as the energy-waves propagate.  In circuits, energy is always EM waves, and the electrons in conductors are the "medium" for the wave-propagation.   The long columns of electrons are not like a drive belt.  They actually are a drive belt, not an analogy.  They are driven to very slowly move in a circle.  At the same time they deliver energy instantly to all parts of that circle.  That's the function of drive-belts, whether made of visible rubber, or made of the invisible "electron-sea" within a metal loop.