Superman throws a light emitting object away from himself fast enough to notice a red-shift. The object passes through a region in which time runs more slowly. From Superman's perspective, does the red-shift change as the object slows or is the light's appearance unaffected?
The object doesn't slow in this case, it speeds up. The redshift is increased. The clock-slowing-down factor is more traditionally called the gravitational potential, and where clocks are slow, this is close to a massive object. In the extreme limit that the light-emitting-object is approaching a black hole, time stops relative to Superman, so that Superman sees the object infinitely redshifted.