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It varies dramatically from one airline to another.
Delta typically (but not always) allows this, but it’s rarely worthwhile on Delta (every now and then there’s a great cash upgrade price, but it’s not terribly common.)
Several airlines will offer relatively cheap upgrades near the day of the flight or allow you to bid for an upgrade if they still have several open seats left in a premium cabin.
However, if the cabin is nearly sold out, then they won’t do that, so it will vary from one day to the next.
Each airline has their own policies for this and whether it’s worthwhile or if it’s available at all varies a lot both by airline and from one flight to another.