The whole point of OFF_CHAIN txns is that you DON'T need to be connected to the network ( ON_CHAIN ).

If I go to the market, and no one has reception, I can still give you valid contracts that you can validate, independently. Phone to phone or whatever.

If one party does connect to the network and tries to spend the txn in an attempt to snatch your funds, then you will have a certain amount of time (the timelock) to logon and claim your rightful coins, and their coins aswell.. :p

Basically - I give StarBucks a valid transaction, that gives them $5. Then I want another coffee, so I give them another txn that gives them $10, both coming from our initial MultiSig setup txn (which is on-chain). They cannot spend both, as they use the same outputs, so they throw the first one away and keep the second (which is more valuable to them).

Neither of these touch the network (require you to be online), until someone wants to cash out.

You don't need to be online to sign / verify / swap a TXN.