The German infantry used one horse-drawn cart (a small one) per infantry platoon in WW2. There was no need to move it into combat itself, but it was a company-level asset and therefore brought quite close to the action.
This kind of load carrying arrangement (=non-combat loads on trucks/APCs) looks still optimal to me unless we're talking about long-range patrols.

For long range patrols over (mostly) wheel-compatible terrain I would have a look at working dogs. They can help as scout dogs (much superior senses), bolster morale, help security at night and at the same time pull a small cart. The latter is possible for soldiers as well, of course. We just don't like to do the job of working animals.
A cart no heavier than what can be lifted over a wall by four men would likely be useful, especially for troops that carry especially heavy single loads (AT and mortar teams, for example).