The Doppler Shift in the Light of Distant Galaxies

We look at the light from a distant galaxy to see how much it shifted. And we use that shift to measure its relative velocity. But cosmologists think about space expanding between us and those galaxies. So instead what's happening to the light is that it's getting stretched out by space expanding. The wavelengths get longer as the light travels through space, so that's why we'd see it red shifted. We're my totally misinforming our public here. You got to get a paleontologist on here to answer that question. I don't think I've ever heard before.