An on-the-fly offer is posted by an EOA (i.e., an Externally-Owned Account, cf. ethereum.org -> Account Types). This is in contrast with a smart offer that is posted by a smart contract.
An on-the-fly offer can be listed on Mangrove but is not equipped with any on-chain logic that executes when the offer is taken.
Whenever an on-the-fly offer is matched by a taker order, the offer sources its liquidity on the EOA.
An on-the-fly offer is not reactive (it has no code) and therefore cannot repost its residual if any. For example, an on-the-fly offer of 1500 DAI (outbound) for 1 wETH (inbound) that is matched by a taker order of 750 DAI for 0.5 wETH will be removed from the book after it has been partially filled.