The only things that aren't hardcoded in your typical secure cryptosystem is your key and a few nonces. In ECC the analogous components to p and g are defined e

The only things that aren't hardcoded in your typical secure cryptosystem is your key and a few nonces. In ECC the analogous components to p and g are defined extremely carefully and certainly wouldn't be changed willy nilly. I'm aware classic DH parameters are more liberal, but changing them for every session seems unnecessary.