Yes. Your forked coins will not move when you move your Bitcoin Balance to a new wallet.
All you have do to preserve any forked coins you want to someday claim is preserve the private key where the coins were held at the time of the fork.
As you have correctly noted, it is best to move all your coins to a new wallet address before attemtping to claim any of the forked coins.
To claim those coins, like Bitcoin Cash, you'd have to download a Bitcoin Cash Wallet, and seed it with the preserved private key.
Seems like an easy task. I'd prefer to do this on my desktop PC for extra security. I've talked about using mobile wallets before, but changed my mind after doing a little digging. The "Pre-Fork" cold storage wallet has a large number of Bitcoins on it, so I wouldn't risk doing these steps on my Android phone.
I have found Electrum wallets for Bitcoin, Bitcoin Cash, and Bitcoin SV. So I should be able to split my coins into new addresses without waiting for the whole Blockchain to sync. I will try doing this to see how it goes. Wish me luck!