When you are using an VFD and motor you can't look at it like a dual voltage motor. You aren't changing the motor wiring at any time while you are running the motor from the drive. Changing the motor wiring on a dual voltage motor is what allows the higher current level at lower voltages.
Below base speed an AC motor is essentially a constant torque device. Motor torque is based on motor current. You cannot go over the motors rated current for any extended period of time. So, assuming that your load is a constant torque load, the motor current will not be significantly affected by speed. Most conveyor application, hoist applications and some surface grinding applications are constant torque.
If your load torque is changing with speed then your motor current will also change with speed. You need to separate the causes, though. Then motor current is not changing due to any spoeed related phenomenon in the motor. It is changing because the load torque is changing. Centrifugal fans and pumps are examples of loads that change with speed.
Then there are center winders. These will appear to exhibit both qualities depending on how you look at it. For any given roll diameter the current does not change with speed. For any given web speed the current does change with roll diameter. The confusing thing is that winder motor speed changes with BOTH line speed and roll diameter. Again, make sure you separate the phenomena before you evaluate.
I have seen cases where the motor current rises very slightly as speed decreases. I believe this has more to do with current measurement filtering than anything else. If you are using at least a decent closed loop vector control or a good quality sensorless vector control the motor current should be relatively constant.
Keith