As with people food, it depends on what we mean by “processed.” Foods that have been traditional for millennia are probably safe, but anything that requires a factory to make is probably not. Simply cooking and canning is probably safe, and the articles I read suggest that dry dog foods with meat and a grain in them are probably okay, but the problem seems to arise with specialty dog foods, such as those containing exotic meats (kangaroo, etc.) or “grain-free” foods that replace the grains with legumes and other vegetables.
I learned with my rats that our animals should not follow trends in human dieting. Rats on a vegetarian diet don’t thrive, for example. On the other hand, they don’t need an all-meat diet, either. What they need is a diet that meets their nutritional needs, and that’s not a matter of ideology, but of known science. Like people, though, rats are hardy and can cope with diverse diets. Proper diet is far more important with chinchillas; deviate a little bit, and they will sicken and die.