For extreme field target where the ranging by focus (and therefore a big honking ridiculous sidewheel) is not necessary, my favorite way to carry my gun is in a padded, zippered case with a shoulder strap. That seems to be the most comfortable and convenient. I do still bring my little cradle to set the gun on during shooting just cuz I don't like it in the dirt.
When shooting regular field target i carry the gun in the cradle primarily because of the sillyness of the giant wheels we attach to our scopes. The huge sidewheels really create a weak point and potential area for scope damage. In my opinion, the gun cradle route protects the fragile sidewheel attachment from accidentally having pressure exerted upon it and breaking the scopes internal mechanism. I cringe to think about the sidewheel getting pressed against my back in the case I use for XFT, or even getting laid down with the weight of the gun on the wheel in a case on the ground. Photos of my simple cradle are in the topic right below this one.
I've not gotten into the cart method, and don't plan to. We carry around and generally manage so much junk/equipment during a field target match that I personally drew the line on this side of pushing a baby carriage through the whole course. I've watched guys struggle to maneuver their carts over and around obstacles and all the bumpalumps that their poor guns are taking through all that, bouncing around and getting violently jostled as they go over uneven terrain/rocks...not for me.