Is it normal after dental implant to move when touch or brushed?

It is not normal for the dental implant to have any movement at all. I would recommend seeing your dentist asap to evaluate the implant. Most commonly, it is the implant abutment and  crown that can loosen and can be easily tightened and recemented in place. However, if your implant is moving, that indicates a failed implant. Your dentist would need to get a xray of the implant to evaluate the problem. 

It is not normal for an implant to have movement. There could be two things that are moving. It culd be the crown itself is moving. This is usually a fairly easy fix. If the implant body is moving it is much more serious. I suggest getting in to visit your dentist ASAP.