you should only be concerned about fee if you hold BTC or ETH, as for the other blockchains? not so much because even at worst time, they'd only require few cents as a network fee, also even if you transfer it right now, big fat chance the fee required will be the same amount in BTC, and once price sky rocketed, your fee deduction will be the same.

maybe the only problem here is if somehow the fee spike up because everyone is moving their crypto in the middle of massive rally, but it depends, since in BTC you'll be charged more or less the same fee whether you send 0.0001 BTC or 10 BTC, if your capital is too low, maybe it's the right time to send it to exchange, but if your capital is high, you shouldn't care.