This took a little hunting around to find, but here's a quick breakdown.
Guerrilla Ops, Supply Raids, Landed UFOs, Retaliation Strikes, and Council missions last for a random amount of time, between 18 and 26 hours. This time is determined when the mission is first created and presented to the player. For Guerrilla Ops, where you have 3 missions to choose from, they will all last for the same amount of time.
Other missions, including Alien Facilities and story missions, do not expire.
A "Point of Interest", which is the name for the rewards generated after completing a mission, will last between 9 and 18 days.
Missions will not expire while the Avenger is in the air, but the only effect this has is that you can't expire a mission you're flying to. If you're flying somewhere other than to a mission site, then missions can expire as soon as you land and aren't interacting with some other dialog.
Anyone interested can find the majority of the relevant code in X2StrategyElement_DefaultMissionSources.uc and XComGameState_MissionSite.uc.
Best Answer
This might be anecdotal, but yes, you can:
I ranked all my squad members to Colonel on the downed Avenger mission, by simply refusing to destroy the signal beacon and farming the incoming assault drops by setting up Overwatch traps.
Took me an hour to do so. After returning to the Avenger I was greeted with multiple rank ups after promoting the soldier (meaning went all the way from Squaddie to Colonel)