Is it permissible for someone who has had a gastric band implanted to perform a certain movement after the dawn (Fajr) adhan to make the pre-dawn meal (suhoor) stuck in the esophagus go down into the stomach, and does that invalidate the fast?

Your fasting is valid. The presence of food at the top of the stomach or esophagus, and then its descent into the stomach, does not invalidate the fast. Similarly, if food reaches the throat and cannot be chewed, it does not invalidate the fast. It is even more so that the fast is not invalidated if the food is below the throat.