There are electrode reactions controlled by electron transfer(slow ones) or by diffusion(fast ones).
Depending on choice of forced electrode potentials, electrolyzer geometry and ion concentration, many reactions can be arranged to be electron-transfer limited or diffusion limited.
If the cathode potential is decreased below its equilibrium potential, the cathode current density starts to exponentially grow (the start of the polarographic step).
With the further decreasing of the cathode potential, the current density does not grow infinitely, but starts to be limited by cation electro-migration, with cations starting to be depleted at the electrode surface.(the end of the polarographic step).
Finally, further decrease of potential leads to the electrode current plateau unless there is yet other electrochemical system able to be electro-reduced.
Electro-deposition of copper is charge-tranfer controlled at low current densities and diffusion controlled at high current densities.
There is direct relation between ion concentration and the current density limit, caused by ion diffusion. In the limiting state, all ions arriving at the cathode due electro-migration, are immediately consumed by the reaction.
There is electrode adjacent electrolyte transition layer where cation concentration of bulk electrolyte starts to drop toward zero when approaching the electrode.
At equilibrium, the cathode reduction and oxidation currents cancel each other to the zero net current.
Note that in context of electrode processes, more important than absolute currents are current densities.
When the electrode potential lower than the equilibrium one is externally applied, the cathode increases reduction current and decreases oxidation current. This decreases the inner cathode potential until the potential difference across the wire causes the current equal to the net redox electrode current.
On the electrolyte side, the net redox current causes cation (or generally reduced object ) depletion, until the extra electrostatic gradient caused by ion displacement forms electromigration current equal to the cathode net redox current.
If the forced potential is too low and the redox net current too high, even almost total depletion is not able to cause high enough electro-migration current. As consequence, the initial net redox current is not sustainable and decreases to match the maximum migration current the electrolyte is able to provide. This is concentration dependent. The wire current adapts to the sustainable net redox current by the inner electrode potential shift toward the external forced potential.