Today, we are firmly convinced of the always the same speed of light in a vacuum as there were convinced in the time of Galileo that the Sun orbits around the Earth.
Like as in the time of Galileo they had no evidence that the Sun orbiting around the Earth, so today we have no measurements that light from a moving source always has the same speed. In the time of Galileo, the orbit of the Sun around the Earth was proven on the basis of hypotheses that came more from emotions than from measurements. Similarly, today we find always the same speed of light.
Objective measurements show otherwise. For example, measuring the wavelength of light from the Sun, measured on the basis of an interferometer, shows the same wavelength of light anywhere on the surface of the Sun. The light from the edge of the Sun that approaches us due to the rotation of the Sun has the same wavelength as the light from the other edge of the Sun that moves away from us. According to Doppler’s law, only the frequency of light changes as a result of the speed of the light source.
Exploring the speed of light, however, is not an end in itself. Proper understanding of light so clears our view of the universe that the universe can be explained without dark matter, that there is no longer a need for dark matter.