Yes, moving aquariums is always a pain. Freshwater tanks are typically easier, but that's more because freshwater animals are very resilient to bad water conditions. When moving saltwater tanks, you usually have to just accept that something will die in the process, even if you're very careful. A disturbance of the sand, rock or filter will change the dynamics of the microbes that keep the water clean, and you get what's called a crash where the bacteria can no longer keep up with the waste. The bacteria will recover, but in the mean time, the cuttles or whatever other animal is in there will suffer and potentially die.
With that in mind, there are some ways around it, especially if it's a fairly simple setup having only the cuttles and some rocks/sand. Fully stocked coral tanks are when it gets pretty near impossible to move. So first off is to have a back up tank to put the cuttles in until you're sure the original tank is ready to take them again. This back up tank can be as simple as a rubbermaid tub with a canister filter, but it has to be the same salinity and temperature and have an active biological filter. To establish the filter, you need a jug of ammonia (unscented) and and a bottle of bacteria, I would recommend Fritz zyme #9. Don't use the cycling bacteria you find at Petco, it's the wrong kind. Once the back-up tank is filled and running, add the recommended dose of bacteria to seed the filter. then add 1 drop of ammonia per gallon every day to feed the bacteria. When you can no longer detect ammonia 12 hours after adding it, then your filter is established. This usually takes 1-2 weeks doing it this way. Bumping the temp up to 90F will maximize bacterial growth rate and shorten the cycling time. You must stop adding ammonia 1 day before the cuttles go in, otherwise the residual ammonia can kill the cuttles.
When you move the cuttles from school, transfer them as fast as possible to the back-up tank, then pull down the original tank. Keep the rocks and sand submerged if possible to prevent microfauna from dying and making you re-cure the rocks. When the new tank is moved and re-filled with water, add more bacteria and ammonia as described above and when you can no longer detect ammonia 12 hours after adding it, then you can house the cuttles in it again.
Make sure the protein skimmer is off for the first day after adding bacteria, otherwise the skimmer will just remove the bacteria.