We have just taken down some tall robinias-they just didn't go with the silver birches! We now have a decent sized border, allbeit in the shade for most of the day and with a leylandii hedge fronted by a wooden fence as a backdrop. we do have some evergreen shrubs that haven't been in long enough to have grown too big, and they could be moved to give themselves more space and also to beautify the fence. When would be the best time to do this please?
Ideally during 'dormant' season..so during winter/early spring...but..you can do it anytime as long as you can ensure the soil around the roots is kept well watered while the plants try establish themselves again.
When you transplant your bushes...puddle the new planting hole with water and plonk the shrub in so it almost float..let the water soak into surrounding soil and to root ball..maybe even repeat the soaking. Give srinkling of bone meal to aid formation of new roots and back fill... ;)
Once the soil is moist deep down it is relatively easy to keep it so. But try to moisted dry soil..not easy and that's why people often fail as usually is is only the top soil that is 'wet'.
You could move them now but I wouldn't. Depends if you are willing to water, water, water all through the summer and autumn.
Goodlife has given excellent advice, except evergreens are best moved in autumn or late winter/early spring. They are likely to die if plonked into a cold, wet soil in January - though it does depend on your local conditions.
If you do move it now, do exactly what Goodlife said, I would fill the hole twice with water and let it drain before putting the shrub in, then a thick mulch on top to help seal in the water - anything would do, even straw or newspaper if you haven't enough compost or well rotted manure.
This article is useful:
http://www.which.co.uk/documents/pdf/moving-shrubs-and-trees-153910.pdf