There are quite a few types of stones/crystals that cats can form. Some are dissolvable and some are not. The most common type of dissolvable stone is struvite and the most common type of non-dissolvable stone is oxalate.
Placing a cat on a dissolution diet like Hill's s/d or c/d can quickly dissolve the stone. To create a state of dissolution, Hill's c/d utilizes controlled minerals, vitamin B6, potassium citrate while maintaining a low sodium diet. Additionally it has DHA from fish oil to help increase renal blood flow. There are other types of prescription dissolution diets but many of them like Royal Canin employs very high levels of salt to create dissolution. It is such a high level of salt, that it induces polyuria/polydipsia (increased urine production due to created of diluted urine). Many of the over the counter diets that say they are for cats with urinary issues often employ this high level of salt. High levels of salt can be damaging to cats if any level of kidney disease is present. If blood and urine tests show that the kidneys are normal, then these other diets can be utilized.
Urine testing to help determine likely stone type is ideal but it isn't always easy. Using a diet like Hill's c/d and repeating radiographs every 2-4 weeks to look for change in stone size can be helpful. If there is not change in size or they continue to increase, then calcium oxalate stone types may be more likely and a diet appropriate to that can be utilized. If the stones shrink, then struvite stones can be assumed and the diet can be used longer term. This plan could be discussed with your vet.
I realize that many commercial veterinary therapeutic diets are not always popular but diets like Hill's c/d have a lot of data and a track record of safety. For situations where we have a known medical issue, this can be a safe and effective method to address these types of issues. This may be a plan to run by your vet.
I hope this helps.