The specific bird in this instance is a kiwi, adapting to life underground as a mole like organism. Would their feathers grow finer and act as sensory hair like structures to "see" while digging? If they were to lose their feathers entirely, would its skin look scaly like the unfeathered parts of a theropod or continue to be fleshy and wrinkled like a plucked chicken? Would the feathers become more compact like that of a mole? Harden into pangolin like scales? What would be useful to this subterranean lifestyle?
There are subterranean birds, the burrowing owl being one. They still fly, however, so this just provides you a start. They are compact (as other owls can be).
Your birds will not have wrinkled skin like a plucked chicken. Your birds will need their skin to be protected, either by scales, fur, or feathers (modified scales). You have latitude on this - Be creative.
If your bird only lives underground, then it will be well-camouflaged, because there will be no selective pressure for colored skin or feathers. It will be some shade of brown. Most exclusively-subterranean animals eventually lose their vision.
All animals need food and mates, and yours will too. If your bird lives only underground, you need to address how it will eat (does it eat seeds?) and how it will find mates. Some birds are monogamous. Some live in groups. These features could be selected for in a subterranean bird.
If your bird also lives above ground, then like the burrowing owl, very few modifications need to occur.