Honestly, dishwasher safe makes your request essentially impossible. None of the options for that are without their own issues. Composite materials are all built using some form or another of adhesive, and those have the same issues as plastic when it comes to swallowing things that aren’t food.
So you’ve got to pick between light materials like plastic and composites you can shove in a dishwasher, or a wood that’s heavier, but only needs quick washing by hand.
So far, nobody has come up with a solution that’s perfect, though wood products constructed with normal wood glues come pretty close in every respect but weight. Even there, while a decent sized wood board is going to be heavy and awkward to wash, at least it isn’t going to slide around.