I don't think I know anyone who has that. I do have neighbors who hand-wash, partly because their dishwasher died and they use the space as a drying rack rather than for a new dishwasher.
Is there a logical place to install one?
However, I also have neighbors who have a rolling dishwasher, it attaches to the sink faucet with an adapter and drains into the sink. Theirs has a kind of cabinet on it, and a fairly nice butcher block top, so it's like a rolling island. They have a fairly tall table as an island and the dishwasher parks under there when not in use. So, this kind of thing might work for you as well.
Thinking about your questions, you'll want to make sure the casters are far apart, and that the base is pretty heavy, both for stability. Other than that, I think something that is easy to clean, to me that means no ornamentation, stainless or solid surfaces, and no unfinished wood, and that can close up for a tidy look.
Good luck!