United is infamous for this, prompting a sound piece of advice: if for some reason you're in first class on a flight scheduled for an A320, don't take a seat in row 3, since odds are high it'll be swapped to an A319 (which only has two rows in F) day of the flight. In theory there's a pecking order for who gets downgraded in that case (first, anyone who moved up to F on a status upgrade, then the paid F fares starting with whoever was latest to check in), but in practice the gate agent just bumps row 3 and calls it a day.

I think the parent poster was in part referring to operational upgrades (very common at British Airways — they will dramatically oversell the Y cabin and op-up people into J/F as needed), which don't necessarily require an equipment swap.

Another option uniquely available to European carriers that you might not have seen in the U.S. is to adjust the number of J/Y seats in an aircraft: on most intra-Europe flights, the "business-class cabin" is the same seats as the coach cabin, but with a blocked middle seat and a movable divider so you know where Y starts.

(Come to think of it, I guess BA will sometimes swap equipment to a flight with fewer classes of service — for example, the LHR-DME route can lose its lie-flat Club World seats overnight sometimes after unplanned equipment swaps — but I don't think BA does this for yield management, because it's costly to them to have to pay refunds to people they've inconvenienced.)

Certainly not in the current airline environment. Even with a perfectly loaded plane, they only make a little money, and you can't plan for a perfectly loaded plane with any kind of confidence without double-booking.

I phrased what I said poorly. They always allow overcommit, but they try to work it out so that on average they efficiently make use of their seats without constantly having to give people giveaways and bad customer experiences.