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The only time there would be "automatic" changes in both axes would be if you were using an autoguider. If the drive was not moving at the correct speed though, I would expect the object would slowly drift out of alignment, but moving the drive in RA should move the object back to its original position, confirming it was caused by incorrect speed.
If the optical axis of the tube is not parallel to the polar axis of the mount it I think it would appear to move out of alignment, however, I don't know how you would check this, but I am sure someone else on the forum would be able to help you out with this if they felt this could be the problem.
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