Distance measurements question in cosmology

I was hoping someone could help me with this conceptual question about distance measurements in cosmology, primarily angular diameter distances. Now, I know that at low redshifts, the luminosity, angular, and comoving distances all recover the same formula. That makes sense as they should if describing the same physical object. What I am confused on is how is it possible that these distance measurements can disagree? For example, take a galaxy at some redshift that is emitting light. If I calculated the angular diameter distance and the luminosity distance, I would receive two different answers for the same physical object at large enough redshift. Am I misinterpreting what these quantities actually represent? It seems most introductory cosmology texts just gloss over the fact that each distance measurement gives a different dependence on redshift and don't agree after some initial point. Why is this?