I hate to tell you this, but it'll probably cost a LOT to ship four cats. I know one woman who shipped her cat from the US to Australia and it cost her around $3000. So for four cats it'll probably be $10k-$12k!

I'm sure that there will be a discount for some aspects of the move since you have more than one animal, but the lion's share of the cost of shipping is the freight itself which is only going to increase with the rising costs of fuel.