Here the problems I see, or have seen.
A couple years ago I was involved with a big two day match with over 100-shooters. The match director allowed competitor to use ATVs and UTVs. I objected intialy based on my perception of human nature to drive/park those things as close to the stages as possible, crowding the limited space we had for rifle staging, shooters, RO's, support staff, etc. I wasn't wrong in my prediction, but here's what I also observed.
The guys pulling carts or wagons generally moved from one stage to another much slower, compared to just walking with gear in hand. The distance between stages was generally no more than 150-feet. This either caused the squad to be waiting for that individual at the next stage, or forcing that person leaving the squad early if they were one of the first shooters to ensure they didn't hold up the squad.
What I also observed with both the Cart/Wagon and ATV/UTV users was they took extra time to pack, load, unpack, unload their equipment and rifles etc. compared to those who just grabbed their gear and walked to the next stage.
Here's where the rub came into play. Because the time it took to load/unload, the guys, especially the groups in UTVs riding together as a mini-squad, would pack up and depart while the remainder of the squad was still shooting. This caused some animosity between those who walked with their gear in hand, and those riding or pulling their equipment. They (the people who walked with gear in hand) were bitter that not only did they have to help out even more with brass collecting, backup spotter, etc., but also didn't have the time afforded them at the next stage to find targets, load magazines, write dope cards etc., In addition to being less rested than those who arrived ahead of them.
I guess because of all those reasons I'm opposed to ATVs and UTVs in precision rifle matches, and also a firm believer that squads need to move from stage to stage as a squad.