oh great topic! Castrations here typically are done standing and then the horse is encouraged to move as much as possible after to drain the incision points (preferably no rolling right away, etc). It is, however, the high desert and we don't have as much mud and things as you do out east so that's an interesting point. I would think keeping them inside/supervised in the first few days would make sense to keep the wounds from getting infected. Otherwise movement, in my experience, is best for the draining/healing process.
I hope all goes well!