Let's say we have two identical spinning flywheels, that have arbitrary geometry, and are made of copper.
Now we apply some heat energy at the center point of flywheel A, causing it to slow down a bit, because some mass was added into it.
And we apply a force, but no torque, at the center of flywheel B, causing it to slow down a bit, because some kinetic mass-energy was added into it. (the flywheel starts to move linearly)
If the energies added are the same, are the new spinning rates the same?
Have we explained the mechanism behind time dilation of moving flywheels?