You can move the currants when the leaves have fallen, and when the ground is suitable, ie not too soggy. Raspberries are also moved when dormant. Autumn fruiting ones should have all the canes cut down to just above ground level. Summer fruiting ones you can risk leaving the canes as cutting them will lose next year's fruit, but do it carefully and replant them straight away.

If they are summer fruiting raspberries I think you still need to cut back the canes that would have fruited next year to about 2 foot to give them a chance to establish themselves. Being moved twice in a short period is gonig to stress both the currants and the rasps.

MG's right of course - I just can't bear to miss a season's fruiting. But it's probably not worth it.