Gravitational blueshift corresponds to the difference between the underlying clock rates at the emitter and at the observer. Which means that the wavelength at the instant of emission is already longer as measured by the observer's clock than the same emission as measured by the emitter's clock. On the other hand, without reference to clock rate differentials, as Einstein first pointed out, energy conservation requires the light beam to gain (kinetic) energy as it "falls" deeper into a gravity well. This causes the wavelength to decrease, so an observer located deeper inside the gravity well measures blueshift. Obviously these two effects are complementary and not cumulative: the resulting gravitational blueshift is not twice the amount of the gravitational time dilation. They are flip sides of a single coin. But can a thought experiment be constructed to enable the two effects to be examined separately? Let's say there is a non-gravitating emitter initially stationary at a certain distance from a massive neutron star. It emits a light beam radially toward the star. Immediately after the light is emitted, a rocket starts towing the star rapidly radially away from the emitter, quickly attaining a constant velocity of .5c. (This is the same rocket that is sometimes employed to tow tethered galaxies.) Then the star is brought to a halt (stationary relative to the emitter) just before the light beam from the emitter arrives. The emitter and the observer on the surface of the star are stationary relative to each other at the time of both emission and reception and no SR redshift is finally observed (although SR time dilation affects the synchronization of the two clocks). From the perspective of clock rate differential, the blueshift finally observed at the star should be exactly the same as if the star hadn't moved. The fact that the emitter finds itself to be relatively higher in the gravity well after emission should have no effect on the light beam, whose characteristics were determined entirely by the emitter's location in the gravity well at the time of emission. From the perspective of the gravitational acceleration experienced by the in-flight photons, the acceleration rate increased more slowly, because the light beam's progress in penetrating deeper into the gradient of the gravity well occurred later than if it hadn't needed to chase the star. It seems to me that the light beam experienced the same average acceleration rate as if it hadn't been chasing the star, but the duration during which the acceleration was applied was significantly longer. Intuitively then it seems that the lightbeam should have accumulated more absolute acceleration effect (energy gain) in the chase scenario. Would more blueshift be finally observed than if the star hadn't moved?