Not unless the token has its own blockchain and acts as its own gas, e.g. EOS, TRX etc.
The only way to move your tokens around for free is by handing out the private key to the wallet, and importing it that way.
How do you mean to import my private key which wallet or where?
Thanx
Some friends has already answered. There is no way for that simply. It may possible with only exchanges (not all of them).
Tokens are using ETH mainnet (or which network they connected. EOS, TRX, NEO...) So this networks needs fee for transaction. Miners process this transaction with fee. And if it possible then miners should have get that tokens. So ETH miners, can earn ETH from their mining activities.
So its not possible even to pay another account transaction fees with a account that Holding ETH ?