Is it normal for a crown to move when you bite down?

It's possible that you are hitting on that tooth too hard. This usually requires a simple adjustment of the crown or the tooth below it. I would recommend seeing your dentist as soon as possible to have your bite checked. I hope this helps.  Click "save" under my name, if you'd like to ask me more questions in the future.  

Hi:This is absolutely not normal.  Please go back to your dentist to check the bite on the tooth.  It can simply be fixed with a quick bite adjustment if the crown itself is not loose.  If you continue to bite on this tooth you might crack the tooth with the crown.  Best,Dr. Maddahi

It is possible that you are hitting the tooth too much and causing slight trauma to the tooth. I suggest going to see your dentist to have him or her check on your bite.

What you're feeling is that the crown is striking the lower tooth that is beneath it too hard. When that happens, it causes the tooth to move out of the way of the bite so that it now feels as though it's moving. It usually requires an adjustment of the crown or the tooth below slightly. See your dentist to have it checked out. 

The movement you are describing is not within the normal range of movement. I suggest scheduling an appointment for you dentist to evaluate the situation.