Not sure what you mean by "moving side to side"? They slowly use their neck muscles to push food down to their stomach, and sometimes when they stretch their
necks out, they lack the strength to hold steady so you might see some "shaking"?...but it sounds like you mean a faster shaking, like a "head wobble" that some
are known for? A video sure would help. How many seconds does the shaking go on for? (counting "one thousand one, one thousand two..."etc)