After using it for my 'everyday' driving, which is MOSTLY 'outside', on dirt trails,since 1986, I'd also have to recommend the Frontier EE cart,for the price and for its sturdiness and longevity(tip: store it INSIDE; it will GREATLY extend its useful life and good appearance!) I have come across SEVERAL that 'look like' the Frontier over the years, but were not NEARLY as light, well-made, or balanced, and as such, were problematic for the HORSE having to pull them!-so I'd recommend being VERY cautious about unbranded 'knock-offs'.
There are several suppliers for wooden or metal wheels that will fit the axles of the EE 'pipe' carts; however, you should be aware that they will give a rougher ride than pneumatic tires. Carts like the Frontier and its knockoffs have only coil springs, and they WILL ride rougher w/o the added 'cushion' of pneumatic tires! You can buy 'no-flat' inserts for the HD tires that come on the Frontier carts(they are NOT 'bike tires', but a heavier type w/ a deeper hub; just don't buy them off the shelf at Walmart-those don't quite fit, and will be noisy-search for the kind made SPECIFICALLY for the wheels/tires that come on the EE carts. I have been told that these inserts also make the ride a bit rougher--but I don't think it would still be as rough as w/ wooden or metal wheels! I have had excellent results w/ just having puncture-resistant tubes and 'goop' put in to my tires at a bike shop, but that's only my personal experience. I have NEVER had a broken spoke; but if you drive over really ROUGH ground, w/ big rocks and tree roots and stumps, etc.,then yes, you *might* truly NEED non-pneumatic wheels!
As for the wheel diameter--larger wheels roll more easily over most terrain; however, if getting wooden or metal wheels, the larger size WILL add significantly to the overall WEIGHT of the vehicle, so it is a real judgement call!The Mini sized Frontier all-metal cart weighs around 85-90 lbs.; wooden OR metal(especially if they are steel)wheels, , are relatively HEAVY, so would add significant weight-I would ask how much each wheel, in each diameter, weighs! A 24" wheel will make your cart 'sit' a couple of inches higher than a 20", and *might* make the shafts too high on the sides of a 31" horse, as there is NO adjustment of shaft height in this type of cart. Perhaps you could ask the company what the LEVEL shaft height is with each wheel diameter, and compare that to the height where you have determined the shafts should 'fall' on each of your horses?(remembering that the shafts should be level to SLIGHTLY inclined upward--never downward-from where they attach to the cart, on this type of cart.)Or, if you go with pneumatic tires/wheels, you could probably buy a set of EACH size, as I see that Frontier now offers a 24" wheel(used to only be able to order miniature horse sized cart w/ 20" wheels; if I were ordering one today, I'd opt for the 24" wheels-but then, I don't drive any horse smaller than 32".
Good luck!
Margo