Sorry, still not getting it. In the "2 of 4" scenario, let's say you signed a transaction on your wallet (eg, Ledger). The next day I sign the transaction using my wallet (eg, Trezor). Does my wallet somehow know you already have signed, and thus can broadcast the transaction? If yes, how?
When you sign the transaction, the wallet will generate you the partially signed transaction which you will need to send to the second signer. Only what the second signer needs is the partially signed transaction from you and then also sign it and broadcast the transaction. This is how multisig wallet works.