One place where I boarded for several years (multiple horses at a time), had conveyor belt strips for mats in the stall, not fastened down in any way. We’d take the mats out once in a while when stripping the stalls completely, if stall floors needed leveling, then put them back in – this was back when shavings were the only bedding option, at least in our area.
They didn’t move around under the weight of horses in their stalls, or when we were picking out stalls (this was a partial self-care barn), or when replacing them after stripping. We never walked on bare mats in your specific type of situation, but (assuming similar conveyor belt material) perhaps they might not require a tremendous amount of pinning down.