I have no experience with this, but my advice (for what it's worth) is to take as much of the rootball as possible (dig as deep and wide as reasonably possible) and keep well shaded while they become established. Watch the soil moisture level carefully, and don't let them get too dry.

Yes, you can transplant hardy cyclamens in June. Actually, that is the best time of the year to do that! I transplanted successfully 6 tubers of hardy cyclamens last summer and they all came back with blooms and foliage in the early Fall. If you dig them up in the summer, you will find out they have little to no roots at all around them. It seems like these tubers go 100% dormant in the summer (at least here in North Texas) and they almost "resurrect" when you least think about them!