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.
All this assumes you already have an open channel with the other party.
I remember someone like achow, nullius or theymos saying that right now, you need that both parties are online, otherwise it wouldn't work, which is a big fail and will never get anywhere when it comes to mainstream adoption
Why would this be such a big problem? When I receive a payment by bank, I need to be online too, to confirm that I received it. More people than ever are online 24/7, and I doubt this will ever go down again.