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It varies a lot.
I travel on United a lot and what I end up usually having to do is book the economy award, then also book a business award (even if the points are crazy high).
As the date approaches, sometimes the business award becomes lower priced then I’ll do that (and cancel the economy award).
To be honest, I’ve had basically zero success while using United mileage plus points for either Swiss, Lufthansa or United to pay for an upgrade.
There are way too many people in line ahead of lowly ole me (people who paid cash already, have some status, etc and are upgrading from a cash ticket).
United Mileage Plus website will outline their specific complex upgrade policies if you want to read at least that carrier.
By the way, I’ve also had zero success upgrading when my original flight was on points with Air France, British Airways, or Lufthansa — could be just me and my luck.
I have though, often had luck upgrading from an original cash ticket on those carriers.
They will email an upgrade offer or they will notify you at the point of check-in.