I move plants all the time, regardless of the "right" time. What works for me is that I pot things up and keep them in my "nursery" area. It gives me a chance to let them settle and recover. Then later I transplant them out when I find a spot, get a chance, or am forced to because the season is ending.
I will say though that when I do did up plants I try to do it in the morning when it is cool and they are at their perkiest. And I water them right away after potting and keep them in a shaded spot for a few days.
I have dug and potted daylilies before and they bloomed in the pot.
I've done iris both ways.Mostly not potted. But not potting them I don't store them in damp cloth or newspaper. I just dig them and they sit in a box without dirt or water until I get to them. I cut all the dead dry leaves off and trim back the others. Once planted they grow new growth rather quickly.
In general plants are much hardier and sturdier than most people think.