I also am still after my 23 year old son and doctors, sometimes pursuing things on his behalf, even though he works in the nephrology field. As you said, as parents we never stop worrying.
Anyway, I asked your question to my older cousin and her son, who had a transplant from her in his early 20s, about our sons' age.
He did have stones before transplant but has not had any after the transplant, which was a long time ago, 20 years ago. So the new kidney in his case definitely stopped the stone production.
My own son has never passed a stone although ultrasounds of his kidneys do show at least one 'floating' stone. We think the reason he has never had problems with stones is that he has been taking potassium citrate since the age of three. My husband unfortunately, completely unrelated, had problems with a kidney stone three years ago. He ended up losing a kidney. He is now also followed by a nephrologist and the only thing the doctor prescribed to him to help avoid future stones, was also potassium citrate supplements.
Hope this helps.