Below is a schematic drawing for a three-electrode cell used for cyclic voltammetry:
Image modified from Wikipedia Voltammetry
This is what I understand with the voltammetry working principle from self-study, please correct me if I am wrong.
At WE, a half-reaction, $\ce{Ox + e- <=> Red}$ is modulated by the potential applied on the WE. Electrochemical equilibrium can be assumed at the surface of WE. When the potential at the WE is increased, WE draws electrons out from the analyte, equilibrium position shifted towards the Ox. When potential decreased, WE 'injects' electrons into the analyte and the equilibrium position shifted towards the Red. In voltammetry, the potential varied continuously, and it is this continuous shifting of equilibrium position (that involves giving or accepting electrons) leads to motion of electrons and hence gives rise to Faradic current.
Using the quoted understanding above (which might be wrong), I couldn't answer the following questions:
When you increase the potential, electrons got drawn into the WE then go towards the CE via external circuit. What will happen when the drawn electrons reach the CE? Will they get injected back into the solution? If yes, what species will accept the electrons?
A typical reversible CV will have an onset potential, beyond which the current increases sharply. If the argument above is correct, even a slight increase of potential should already lead to shifting of equilibrium and hence Faradic current, and so there should be no onset but rather a direct increase in current as soon as potential is applied.
Say you keep changing the potential applied, the equilibrium position will be shifted continuously and hence Faradic current observed. What if you stop changing the potential at one point, but hold the overpotential at a non-zero constant? What will be observed? 0 current?
When there is no redox-active species in the electrolyte, current will increase very slightly when potential increased, and that is due to non-Faradic current (capacitive current), due to ions accumulating at WE surface. I don't see how ions accumulating at WE surface will lead to current flowing between WE and CE and hence gives rise to non-zero current.