Its depends on the state... when I lived in nevada you HAD to have a brand inspection and a current coggins and health cert just to drive 10 miles down the road... The highway patrol could and would pull you over JUST to check your paperwork.. no paper work+plus a high on authority po-po = a potential for your horse to be confiscated until the proper paper work could be provided..
It may be different now because there was kind of an uproar about it, but that's how it was..