You can either Sweep (which will create an onchain transaction that sends the BCH from the old "btc" addresses to new BCH addresses in your BCH wallet)... or you can Import the keys into a BCH wallet (use the "Import Bitcoin addresses or private keys" option when creating the wallet in ElectronCash).
It depends what you want to do... if you want to HODL your BCH, then Sweep is the better option as you're moving the coins from the "exposed" keys. If you're just wanting to dump the BCH... importing the keys and then just sending them to an Exchange is probably "cheaper" and faster (saves paying and waiting for the sweep transaction)
can't you just import your keys into an offline ElectronCash wallet as a cold wallet option for hodling?
I don't know how to configure an "offline" ElectronCash wallet. It probably can be done. After extracting the coins you just send them to your hardware wallet. Wash repeat with next forked coin.
Or you can add a password to the ElectronCash wallet and keep the coins there until you figure out what to do with them.