You can run the 50Hz transformer on a 60Hz supply. Now since the frequency has changed the only variable thing in the existing transformer that can change (assuming simple one input and one output) is the flux density. In its simple form we are operating the transformer (60Hz/50Hz = 1.2) at 20% lower flux density.
If the things were reversed the flux density would be 20% higher and may saturate depending on design. You can even run the transformer at 400Hz providing that the voltages are reduced (GOSS can be operated at maximum 1kHz).
Operating 60Hz motors, switchgear etc of the 50Hz designed transformer will have no difference than operating form the transformer designed for 60Hz only.
If the transformer is designed to operate at 50Hz and then you operate it at 60Hz.....How is it possible that you will increase the transformer losses? You are lowering the flux density making the Iron losses lower and not higher.
The only reason I can see why we lower flux density in large transformers is to keep them quiet. So again the lowered flux density would theoretically make the transformer less audible even if you are applying 60Hz to it. If the transformer core is constructed properly it should be nice and quiet.
Now operating motors from 50Hz to 60Hz and vice versa is a completely different topic and I think there is lot of articles confusing the two. But by using a solid state frequency converter, you can easily change from 50Hz to 60Hz, or 60Hz to 50Hz power supply.