Cosmological redshift causes wavelengths of a distant object to stretch by a factor $1/(1-Hr/c)$ where H is the Hubble constant, r is distance, and c is the speed of light. Consequently the received power density drops by $1-Hr/c$ from that factor alone. Combined with the inverse square law, this would suggest the apparent brightness dropping as $(1-Hr/c)/r^2$.
However, the Cosmic Microwave Background (CMB) has maintained its characteristic black body radiation density with just a drop in temperature by a factor of (1-Hr/c). Since black-body radiation goes as the fourth power of temperature, it suggests that the apparent brightness of a stellar black-body including the expansion of the universe should go as $(1-Hr/c)^4/r^2$
So the question boils down to this: Does a distant red-shifted stellar object look like a slightly cooler black body consistent with its actual diameter, or does it appear brighter than that?
[Edit] Just to capture @benrg's answer which I marked correct: A red-shifted black body looks cooler by a factor $(1-Hr/c)$ and it's apparent brightness is less by a factor $(1-Hr/c)^4$ consistent with its temperature. This is true whether the redshift is due to simple velocity or cosmological expansion.