Question - When is the best time for moving koi carp from one pond to another

I would say early fall. If not then, I would say mid-spring. Really, they can be moved at any time as long as the pond is not frozen and precautions are taken. You want to make sure there's not too much of a temperature difference, the fish both in the pond and going in are healthy, and the pond's filtration and good bacteria can take the extra load. It's easiest to see koi in a pond in late winter so if the pond is highly vegetated (that they are leaving from), it may make sense to move them then.