Over simplified, but basically the magnesium ( Epsom Salts or magnesium citrate) allow the ducts to relax enough to let even large stones through. Not eating any fat all day before flushing allows the bile to build up which allows for more bile to 'push' the stones through the relaxed ducts. Think of it as having a marble in a water hose. A warm hose is much more relaxed and pliable than a cold one and a full water flow rather than a trickle pushes the marble out easily.
If you're very worried about large 'stones' getting stuck, try taking Gold Coin Grass and Chanca Piedra (also known as "stone breaker" for good reason) and possibly drinking a glass or two of *room temperature* apple juice for a month or more before doing your first flush. (If you have problems with the Sugar in apple juice, try taking malic acid.) The herbs help soften and even break down many 'stones' while the malic acid from the apple juice is a softening agent as well. During that time, check into some liver supporting herbal teas as well to help strengthen and prepare your liver.
A 'stuck stone' from Liver Flushing is actually very, *very* rare. Usually, if one feels a 'stone' somehow did manage to get 'stuck', an extra does of Epsom Salts helps. It's also fine to do another 'mini flush' of Epsom Salts followed by the olive oil / grapefruit juice if you feel it's needed.