It would probably depend how far you are moving and what the access is to both (old and new) fields. The livestock hauliers may have rather large lorries. You would have to have them in a small accessible pen ready to load, but only one journey would be less stressful for both you and sheep.
Chooks in boxes safely packed into the trailer I think. You can get poultry carrier boxes, they are stackable and plastic mesh, so no issues with ventilation in a horse trailer.