That is how mobile generators operate in 99% of situations. "Mobile" is referring to how the generator can be transported to different locations to be used where power is needed.
No, “mobile generators” are not constantly powered at one location.
Typically they are unused in storage and brought to location without power. It could be a warehouse with sensitive equipment (freezers etc), a hospital in need of extra power etc.
This is clearly a case of abuse and people should go to jail for wrecking the planet.
I work in the energy sector and I’ve done routine maintenance on backup generators, including work on the diesel engines.