In almost all cases _total_ losses will increase as the load increases.
However the various internal losses will change in relative proportion.
My gut feeling is that the 'core' losses (magnetizing and eddy current) would _decrease_, since the input voltage for magnetizing would drop slightly. But this is from experience with induction motor design; the various loss aspects mentioned above might have the opposite effect on core loss.
At the same time the load conduction losses would increase in proportion to the square of the current. I expect lower losses in the steel and higher losses in the copper as the load increases.
-Jon
