I will just give some options from experience and from what others have done.
Where I live, the temps get hotter and the summers are typically drier than NC but much of it is the same growing zone as here. Potting them would work since you could relocate them in the shade when you move but it is lots of work and I probably would not go thru that. I have received daylilies in June a couple of times since I ordered from northern growers and they could not get them to me any sooner. I lost some plants both times even with the best of care.
I understand your moving the plants, I would want mine if I moved. I never want to have to do that since I have thousands of daylilies and 500 iris.
I know some southern daylily growers who move plants in the summer and make it work. Lots of water and maybe some mulch could keep them going just fine.
I would dig the daylilies as close to the move as possible and cut them completely back. I would temporarily locate them in mostly shade for the summer. I have just dug trenches where I could get their roots in the ground quickly. As you get to fall or late summer cool down they can be put in their permanent beds.
The iris can be dug any time and planted any time. They should be fine. I dug lots of iris I was relocating to new beds one year. Months later in the fall, I found a bag of iris that I had somehow dropped or failed to get when planting that was lying in a full sun area. That year was our hottest on record with 63 days over 100 degrees. When I found the iris it was brown dry and dead looking. The rizomes did not look that bad so I just stuck them in the ground and over half of them started growing.
Sounds like lots of work, but then I have a hoop house with 5000 seedlings growing in it and they all need planted out in the garden in April.
I am sure New Hampshire is a wonderful place but North Carolina will be a much different area for gardening and it depends on what kind of place and soil you have.
I hope your plants will take the move and survive.
