I agree with charlie that the physics of electrical power distribution makes the changes to the cable carrying the power over the last few feet should not affect the way a piece of equipment operates, at least as long as the original cable wasn't significantly undersized or damaged. The only thing that last bit of cord needs to have is a conductor that is properly sized for the peak power consumption of the piece of equipment.

That does lead to a few "stating the obvious" observations. One, if the cord you are starting out with is undersized, then replacing it may offer benefits - it may actually allow a component to receive the full current flow that it needs. Carrying that a step farther is another observation, which relates to the post that started this thread. If the circuit that a cord is plugged into is overloaded or fed from a panel that is overloaded or otherwise restricted (corrosion on conductors that limits current flow, as an extreme example), then you are in much the same situation as you would be with a skinny little power cord connected to a properly sized and installed circuit.