Speaking from an AMHA standpoint (R is probably similar but I have never done it there) in multi hitch they are looking for evenness of pace and working together as a team. They should be well behaved and responsive. The overall appearance should be as proportional as possible and matching is also considered but not so much in color as in size.
In light harness you will walk, collected trot and working trot. In draft harnes you will walk and collected trot. You enter the ring just like a regular class and work both ways. You will then line up at one end of the arena where the judges will inspect you and ask you to back. After that you will, in turn, move to the other end of the arena where you will present to the judge (this can be as simple as stopping and nodding to the judge or as elaborate as a full ADS salute) and then execute a figure 8.
Antique Vehicle is a show class where the only requirement is that the vehicle have wooden wheels. It is done as a regular driving class with walk, collected trot, working trot, line up and back. Five years ago you would see Jerald carts with wooden wheels at the World Show. Within the past several years the class has become popular and quite a contest with pairs and four in hands pulling elaborate vehicles dominating the class.
Here are a couple of our Antique Vehicle entries from the past several years.
Single
Pair
Four