I dont think the kidney stone would have moved like that. But since you developed one in the first place its not that unlikely for a new one to have grown on the opposite side.
I agree with pixiepie they cannot move from one side to another. you probably passed the one on the right without knowing and new one formed on the left. Also a kidney infection will have the same symptoms as a stone, the only reason you have pain is because of blocked urine flow and swelling of the kidney which happens in infection and stones. If you haven't already you should really be seen by a doctor and have a CT scan done to tell which it is. Also they can measure the size of your affected kidney to see if the stone needs to be removed or if you need antibiotics.