Let's say the anode is 1/3 copper, 1/3 silver, and 1/3 gold. The copper will plate out until none of it is on the surface, at which point the silver will start to plate out.
If neither copper or silver are available, the gold will start to plate out.
You wouldn't really end up with a pure copper cathode; you'd end up with something that's mostly copper but with some amount of silver and a (smaller) amount of gold.
With the right setup they can bring a sheet of metal that's 99.99% pure to 99.99995% pure in one go.
When you refine metals via electrolysis, there are two reactions to consider.
You have the anode, which you "pull" metal off of; the reaction that occurs here would look something like:
Cu(s) -> Cu2 (aq) 2e-
And the cathode, where metal plates on:
Cu2 (aq) 2e- -> Cu(s)
If your anode is made up of a number of different metals, some metals will be "pulled into" solution more readily than others. These metals are often called ignoble or "less noble". If you consider the electromotive scale, elements with a higher electrode potential are more noble; elements with a lower electrode potential are less noble.
In this case, the anode is made up of gold, silver, and copper. The electrode potentials of these metals differ.
- The potential of Cu(II) is 0.3419V
- The potential of Ag(I) is 0.7996V
- The potential of Au(III) is 1.4980V
As you can see, copper is the least noble element and will be "pulled off" of the anode first. To simplify things, let's assume that all of the copper will enter solution before either silver or gold will.
Now we consider the other half-reaction, where metal plates onto the cathode.
More noble elements preferentially "plate out", leaving the solution and forming a solid on the cathode surface. Recall that we have been putting copper into solution; if this is the only element in solution it will plate out, leaving us with a cathode covered in pure copper.
What happens when we deplete the anode of all copper?
Silver begins to dissolve into solution and plates onto the cathode! Once all of the silver plates out, you're left with nothing but gold.
Of course, it isn't this simple. The gold and silver will form slimes on the anode surface; some will enter solution despite being more noble than copper and therefore plate out before the copper can (resulting in an impure product). Dendrites will form on the cathode surface, causing different voltage potentials in the cell and varying current flows, and (given enough time) will reach from the cathode to the anode.
It's a complicated process.