I am building a world with islands floating in air, and I'm looking for solutions to loss of water from runoff/erosion (think rivers emptying off the side of the islands). The islands vary in size, with the smallest around the size of Manhattan, and the largest about half the size of Iceland.
A few limitations: sunlight does not reach the surface below the islands, and nothing comes from the surface to the islands; the islands may as well be floating in a void in terms of how the surface affects them.
Not all islands are inhabited, and those that are developed their science and technology while on those islands, so while a more efficient modern solution is possible, there has to be a reason life didn't die off before it was reached.
Given the limitations, is there any way to avoid the loss of water through runoff? Solutions involving science, magic, or both are equally welcome!