Note that the universal wallet format does not need to equal the internal wallet format used by the software. Wallets can store their internal database in any way that is efficient (and secure) for them as long as there is a way to import/export the universal format.

Bitcoin Core supports the `dumpwallet` and `importwallet` RPC commands for going from/to an interchange wallet format. This is basically just a text file with a list of private keys with metadata.

BTW: in general, unless you have a very good reason to do otherwise, I'd recommend to generate new keys and transfer your coins when you switch to a new wallet. This is safer and better supported.