The good news is that the unknown Heuchera at my folks' house has done well over the years...and there are some smaller ones in the shade of a hydrangea that I can move to my place. So...when would be best to do so?

This pic shows part of the Huechera complex they have. The flowers are a small creamy white and draw LOADS of smaller pollinators...really productive. Those in the shade did not flower.

In my experience, any time when it's not super hot. I move mine all the time and the biggest thing of importance is to take your shovel and grab the biggest clump of soil as you can, so the roots are disturbed as little as possible. Get it into the new location quickly and give it lots of water. Should be fine. If you have to move them long distance, maybe pop that big shovel full into a plastic garbage bag, tie some string around the dirt clump to keep it as tight and intact as possible and get it to the new location as fast as possible. If it's going to be in the bag for a few days, water the heck out of the plant an hour before digging it out, let the roots soak up that water really well, then dig it out and wrap the dirt ball in a plastic bag to keep it all intact around the roots. It should be fine that way for a few days. If it's going to be more than a few days, you might as well pot it up into a plant pot with lots of soil around the roots. Then it can last indefinitely. Good luck!

You can do it any time as long as it is not to hot, but Springtime is the best time to do it.

It's still VERY hot here...and humid. Is transplant shock more risky in the warmer weather? I have a better idea about the size and location of the plants now.

I would not transplant while it is hot and humid. Wait till things cool of or till Spring.