The analyzer will probably function as a ground plane for the antenna.
When you pick it up/move it, could it be that you influence (extend) the ground plane with your hand/body and that this effects the measurements?
I am no expert but I would not be surprised when the analyzer itself would influence the antenna measurements, because of the ground plane effect which may skew the results.
If so then adding a cable between antenna and analyzer will probably reduce the effect (but possibly introduce other side effects).
In other words, how reliable is the measurement when the antenna is directly mounted on the analyzer?
How representative is the analyzer as ground plane for real world situations, and would it maybe be better to add a cable (short enough to be of minimal impact but long enough to reduce analyzer ground plane effects) between analyzer and antenna to test the antenna in a (from ground plane perspective) more representative situation?