The Room
Consider a room that is a perfect square and has a light source in the exact center. The assumption is made that the time gradient (TG) associated with the room has the same density everywhere in the room. If the room were not in motion it would then be expected that a flash of light from the stationary source would be seen simultaneously at the exact centers of each wall (roof and floor will be considered as top and bottom walls). If the room were moving the same thing would be expected to happen. Why would this be the case?
The emitted light starts as a spherical wave leaving the source with its speed controlled by the source RF. The photons also retain the velocity imparted by the source/room. After leaving the source RF, the forward (radial) speed of the light in all directions relative to the room is governed by the clock rate in the room RF. The photons can be assumed to retain the source velocity, which is the same as the room velocity. Thus the light would be expected to continue expanding as a spherical wave centered at the point where the origin flashed. Since this spherical wave must be moving with the room, it will reach the center of each wall at the same time. As far as any observer in the room is concerned, the light behaves as though the room were stationary.
Now consider a light source that is moving within the room with some velocity in the same direction as the room velocity relative to the room. When the moving source is in the exact center of the room, it emits a flash of light. That light starts out as a spherical wave, which has the velocity of the source added to it. When the light leaves the source RF, the room RF regulates the radial speed of the light so that it continues to travel at the same forward speed relative to the room in all directions from the point of origin of the light at the center of the room. Thus, relative to the center of the room, it should appear as a spherical wave that is expanding out from the point of origin at the center of the room (and not centered around the moving origin[1]). The first contact of the light with the walls would be in the center of each wall.
Since the photons would retain the velocity of the source (which is greater than that of the room) unless there is a conflict with the forward speed of the photons as determined by the clock rate within the room, it would appear that the photons would shift toward the front of the spherical wave. In this case, the photons that were emitted perpendicular to the path of the source would not be the ones to initially strike the walls. The photons that initially strike the walls would be those emitted in a somewhat backward direction. This overall effect is called the photon shift, and is discussed in more detail in a separate sub-section. It appears to be the photon shift together with the effect of the room TG on the photons that keep the wave front centered at the point of emission rather than following along with the source.
[1] According to special relativity, the light actually travels in a sphere relative to both the center of the room and the moving source due to the way that space is perceived from both positions. A pictorial demonstration and explanation of this is shown at http://casa.colorado.edu/~ajsh/sr/paradox.html. This is not perceived to be the case in the theoretical approach discussed here.