Before mentioning any comments, what is the reflow configuration, paste which you are using and also the location of BGA in the PCB i.e in the center or on the PCB edge?

Are you confirm the BGA mounting at pre reflow?

I would check the level of the oven, and the velocity of the convection fans.

Before I got too crazy, I would rotate the PCB 180 degrees and see if that helps.

This is an interesting problem because BGA devices are supposed to self center when the solder melts.

If the board is warping at temperature or the rails of the oven are not flat, would be a place to start looking.

I would also look at how long the board is at temperature for. Perhaps a longer or shorter time would work.

If all else fails you could always lock down the corner of the device with a surface mount adhesive.

Without seeing the board, the types of components being loaded, board finish, and the paste type being used, the following are possible causes:

Excessive surface tension under the part could be the cause. To address this, try reducing the volume of paste the stencil is depositing by either reducing the stencil thickness, or reducing the size and or shape of the aperture.

Oven profile could also be a factor. Check to be sure the zones are uniform. Inconsistent heating zones could be causing the shifting.

Finally, large ground planes or larger components on the board could be acting as a heat sink or heat shield, could be contributing to the problem.

Movement of a component could be caused by elevated mechanical vibrations stemming from a questionable conveyor system or blower motor.

Other sub-systems to explore would be sources of impinging gas, such as outputs of convection plenums, cooling tubes, recirculation returns or gas distribution plumbing in both the heating or cooling regions of the reflow system.

Uneven heating or especially cooling can cause dissimilar forces that can shift a component, this is frequently witnessed with passives but larger devices are not immune from this phenomenon.

A quick test to eliminate the heating & cooling variables would be to duplicate the process with a few identical samples processed through the reflow system while set-up to operate at room temperature.

If the component continued to shift, chances are its oven related. If it didn't move, then it may be thermal related such as device uniformity, heating rates and cooling rates. I'd also double check pad-package geometries to be safe