How not to do it
A technique that was commonly used in the past to induce colonies to replace their queens by supersedure was to damage or cut off a leg of the queen. Although I knew about it from older beekeepers when I started beekeeping, I have never done it myself. I am happy to clip the wings of a queen because I don't believe that will cause pain, but I think a leg is different.
Supersedure should naturally happen towards the end of the season, but it has become common for colonies to supersede their queens during the summer, which is one of the queen problems modern beekeepers have to deal with. This often results in the colony swarming with the old queen, leaving what was intended to be the supersedure queen to take over the colony. For this reason I suspect the technique of cutting a queens leg off was probably done towards the end of the season.
As with many of the tricks of the past they worked, but the beekeepers didn't know why. The practise had largely died out before we understood that a queen produces footprint pheromone that is spread on the surface of the comb when she walks on it. Obviously a queen with a damaged or missing leg would spread less pheromone, so inducing supersedure.
Although I don't condone this method, it did allow the old queen to lay fully until she was replaced by her daughter, so there was no brood break.