moving across
This sentence says that the road trip included a transverse route. Since it's a road trip, it will be accomplished by a moving vehicle. Also, we know that New York City and the Grand Canyon are on almost opposite sides of the USA. This means looking across and writing across are definitely wrong. Could the vehicle be leading another vehicle across? Yes, but we don't have any indication of another vehicle or something that needs to be lead. This only leaves moving across as the only option.