In fact, as a diuretic, one would think that caffeinated beverages would have an opposite effect; kidney stones crystallize when urine is in concentration.
Actually, the diuretic effect helps contribute to kidney stones. You are right that caffeine isn't an actual component of stones, and that concentrated urine is a factor in stone formation.
Nevertheless, the frequent urination associated with it keeps the kidneys so flushed of water that insoluble calcium oxalate (component of most kidney stones) starts to collect in the kidneys. What results is a stone that increases in size as more and more insoluble material is precipitated.
Calcium oxalate is insoluble no matter what but when it enters the kidneys in a large volume of water, the particles remain small enough to pass right through with the water and be urinated out. This is a continual process so the urine concentration comes into play-most patients are encouraged to drink a half-gallon + of water throughout each day to keep liquid flowing through the kidneys and not allow insoluble matter to collect. When very little liquid remains in the kidneys at any given time (as with diuretics), the small particles collect in the kidney and with the aid of the warm environment, lump together to form a larger crystal mass (kidney stone). Because diuretics continue to stimulate urination even when little liquid is passing through the kidneys, this can soon lead to the precipitation and buildup of calcium oxalate.