Rules vary state to state, so check your destination state's regulations BEFORE the hunt — not after the kill. Most state wildlife agency websites have one-page summaries of carcass transport rules for hunters bringing deer in from out of state.
Processing in the harvest state is the cleanest path: debone in the field or at a processor, leave the bones and inedible parts in-zone, and travel with cuts only. State disease risk pages link to current rules.