I admit that it is also always for me a period of crisis and loss. I do not know how to answer this question correctly.
for the most part, I would say it's springtime (even with pci iris, I had fewer losses doing it in the spring).
for the irises of the Oncocyclus section, it is probably August / September (well I think) before they make their new roots.
whatever the species, the principle is probably to transplant them just before they start their new growth.
however, now I am still waiting for at least the 2nd or 3rd year of growth.
if someone has other ideas, I'm interested.