Yes there is a way to do that. It is done by scripts and a smart contract, script feeds the address list to smart contract in bulk packages, and smart contract sends them with the standard send command (looped).
The avarage cost of airdropping tokens is 0.1ETH per 1000 addresses ( I'm talking about gas cost only ). So for your examplary 5000 addresses, you'dpay 0.5ETH gas fee.
Thanks for the info about the gas cost. I can see, from the Twitter, that you are very knowledgeable about all the airdrops happening!
Do you know of any example smart contracts / scripts used? All of the tokens mentioned on your Twitter must be using such code.
Thanks.