@the OP:
Is there any information about the airdrops on EOS chain? Should we move the ERC20 out of Kraken so that we're eligible for the new tokens or the exchange will do it for us?
Thanks in advance for your answer.
If they didn't write a blog post or tweet about this then I suggest having full control over your private keys. Exchanges usually only support forks and airdrops of coins they are going to list in their platform.