Relayers
After the deposit, acts go through a relayer, so no wallet of yours signs them. The relayer takes your proof and its external data, decrypts its own fee note to check the fee, and submits as fee payer. Its fee is a note to its own Omnibus address, in the asset that moved, so even the fee doesn't show which asset it was.
Submit it yourself
When the relayer doesn't answer or turns an act down, the account offers to submit it yourself. The same send, delivery, swap or statement is proved again with no relayer fee, and any connected wallet with SOL signs and sends it as fee payer, paying the network fee and rent. It can be a different wallet from the one that opened the account. The receipt is the same as a relayed one, and the act's original notes are kept, so it can't be paid twice.
A proof made against a recent root is accepted while that root is in the ring.