If the tank is fully cycled, then moving your livestock from the old to the new would be much the same as moving it from a dealers tank to your own. Only less stressful becuase your water is going to be simalar in both tanks(having come from the same source) and there's no transit time.
Easiest and safest way to do the transport would be to get a styrafoam box from your LFS, or use a bucket if that's not feasible, and then put your livestock in the box, w/ some of the original tank water, and slowly drip new tank water into the box untill the water amount has doubled. Then transfer the livestock to the new tank.
If you have very delicate livestock, you can double the water amount, then take some out, double it again, and so on, untill the water in the box is fully the new tank water, allowing for a long aclimation time that should be stress free(as long as the water temp won't drop during the aclimation)
If you are transfering a lot of livestock, I've always found seachems Stablitly to do a great job of keeping the water param's stable till the bacteria can catch up with the extra bioload.