this advice is not from moving experience, but 6 hours in a carrier is not really that bad. Think about if they were at a grooming facility or vet's office for a procedure. They could be in a cage even longer. So as long as it is not a tiny cage in comparison to their size.........
I may suggest witholding food and water from about 8 p.m. the night before (assuming travelling around 8 a.m.) so they may not have a full bladder or bowels for the trip. This is the same thing the vet will ask you to do if the animal has surgery the next day, so it is not harmful. And I think that an added benefit to this is that if the cat is somewhat stressed he may not want to eat right away in his new home, but if he hasn't eaten since the night before, hunger may win out over stress. Good luck