Let’s suppose you have "tight" tuning of KP>300 or so and KD>2000 or so. This is just an example of slightly tight tuning, but not too high. The higher the numbers, the faster motor current will rise under a given increase in Position Error. Collectively, with the above mentioned facts about "E" maximum allowed Position Error, the current may rise much faster. It is best to ratio acceptable tuning values with good Position Error values so as to maintain the lowest running Position Error with the lowest value of "E" possible. The ironic thing here is that usually to get decrease following error implies increasing tuning. This is true, but for example: KV (velocity feed forward) and KA (acceleration feed forward) are better means to achieve this goal.
These tuning values lower Position Error while moving without increasing motor current because they shift the motor position command forward in the trajectory for the entire move, compared to during the dynamics of changes in moves. As a result, you get lower peak currents in the motor.