Don't know about other states, but in Wyoming when we transport any livestock out of the county, they have to have a brand inspection. This applies even if retaining ownership to go to owned or leased pasture or to be placed in a feedlot or to go to a show. Livestock we frequently haul (horses, show cattle) we can obtain a lifetime brand inspection paper.
All livestock leaving the state have to have a brand inspection for the change of ownership and health papers if the buyer's state requires them, which most, if not all states do. Unbranded livestock will show "No Brand" or "NB" or something like that to indicate that they are slick.
Incoming livestock only need a brand inspection if the state from which they originated requires a brand inspection, but in all cases need health papers. For instance, during the NWSS at Denver, if cattle come from a non-brand inspection state, they need health papers. When they are purchased and leave Colorado, they need a brand inspection to leave the state as well as health papers to enter the state of their new home.
For cattle coming to Wyoming directly from a non-brand inspection state, all that is needed is health papers. I always want a bill of sale from the seller for taxes.