I've transplanted it spring, fall and midsummer. As long as you keep it watered it should survive.

I just divided and transplanted mine on Monday night. It slumped a little but it's been pretty hot here. We're getting like late July and August temps in late May/early June! I keep watering it every night and it seems to be doing well.

But it is [possible for astilbe to be in too much shade...I had some that didn't bloom last year...I guess because a bush which shaded them grew. I trimmed it and they have buds.
I would transplant after they bloom...so as not to lose this year's bloom.
Linda C