I suppose lots of people have their own way of doing it, but my method includes giving the plants a thorough watering first as this will reduce shock of being disturbed.
Next, prepare the planting hole - make it bigger than the rootball of the plant, add some compost (optional - it can help for roots to be able to take up nutrients not long after planting. Also consider how fertile the soil is - you could add some fertiliser such as BFB or Growmore into the hole as well) If the soil is dry put some water in. You can put lime in the planting hole if there is risk of clubroot or if the soil is on the acidic side.
Carefully loosen the soil round the plant and down - use a fork rather than a trowel in case the roots have spread out wider than you thought, and scoop up. Avoid the soil round the roots being disturbed if you can, and transfer to the planting hole.
Brassicas can be planted deeper than the original soil level. Back fill with the soil and compress round the soil (depending on the soil, using the heel of your boot may be necessary) - this helps to prevent rocking in windy conditions, as brassicas are not fond of loose soil round the roots. Presuming these are for sprouts next spring (?) you might want to consider staking for additional sturdy support as the plants get bigger and have to endure winter and strong winds.
Finally, give the plants another thorough watering - if you did damage any roots, the water will keep the plants from wilting so quickly. If you transplant on a hot day or windy day, this will become more evident but normally they recover well afterwards.
Checked growth can happen if roots are damaged or disturbed, but I'd say brassicas generally are a lot hardier than say, tender summer crops so should be alright.