50Hz transformer running on 60Hz power supply

Yes. Increasing the frequency is allowed from 50 to 60 hertz on most American made industrial type transformers. You are lessening the burden on the core steel.

The secondary response:
Voltage and Currents are separate issues from a frequency change of 50 Hz to 60 Hz. In transformer design, we don't really care what the frequency is as long as the turns ratio on the core steel won't generate too much heat. You are fine there. The voltage output will be of the same ratio of what you put in. There may be lower overall loss, but you are still looking at the turns ratio. Look at your transformer nameplate.

As for the other equipment: You need to operate each individual component based on the spec sheet/operating voltages for those devices. A lot/most equipment is designed to run for either 50/60hz and with a tolerance for input voltage/current, but it is individual to each component. Double check this.

50 Hz transformer can be used for 60 Hz operation, but reverse operation (i.e 60 Hz Xfrmr on 50 Hz operation) will result in damaging the transformer.
Theory goes,
Each magnetic material which could be used in a transformer's core has a limit on how many flux lines it can handle.
The lower the operating frequency the more flux lines that are generated. Operating a transformer at 50Hz generates 20% more flux lines than at 60Hz. As the number of flux lines approaches the magnetic material's limit, the heat in both the transformer's core and its coil wires increases, and under certain circumstances, unpredictably so. This can result in a transformer that exceeds safe temperature levels. Therefore, a transformer designed to run at 50Hz will simply run cooler at 60Hz. But one designed only for 60Hz may overheat if subjected to 50Hz.

i.e. from 60Hz to 50Hz, your transformer impedance will drop by 20%, therefore increasing the primary current and could saturate the core by having a higher flux. If saturation occurs, then the current flow in the primary can increase even higher and overheat.

Usually, saturation will not happen on the lower end of the loading: that is if you do not load the transformer to its max rated watts. 80% less than the max load will still be safe, even though it will run hotter when compared to the same load on 60 Hz. Keep it at 70% better.

From the voltage ratio transformed, the same ratio will remain between the primary and secondary since this depends on the number of turns in each winding.

However, you can use the transformer on 50 Hz if the voltage on 50Hz is lower than the one on 60 Hz in the range of 20% less.ex: if 230V at 60Hz, then 190V at 50Hz. or 127V at 60Hz then 105V at 50Hz.