Position-wise, mushrooms can spread by (slowly) moving. They can also move by detaching and allowing water flow to transport them to a different area of the tank before they take hold again.
Numerically-wise, mushrooms can spread by asexual reproduction either by self-dividing (I guess the term there is fission), or from any cellular fragments that the mushroom leaves behind or which may result from damage. I've even seen mushrooms develop from ejected mesenterial filaments (which the water flow carried off to a different sector of the tank.
Rift Lake and Reef tanks.