Contacts moving is a common problem for contacts wearer. According to ophthalmologist, it is because that the corneal base curve does not fit that of your cornea. If the contact lens is too tight, it cannot move properly. If it is too loose, it will move in your eyes and result in blurred vision.
Besides, if wear it inside-out, it can also move and the eye will feel uncomfortable. Take it out, put it on the top of index finger and confirm whether it is inside-out.
Last but not least, a contact lens that was worn for a long period of time can still result in movement. For an old contacts, some secretion could accumulate on the lens. If you cannot clear it thoroughly, bit by bit the lens would crack due to the working of those secretion.
So, for the first situation, you must go back to see your doctor and reconfirm base curve of your eyes. For the second one, wear it properly. For the last one, replace a new one.
Good luck!