After a Tummy tuck procedure, will the fat cells travel to other parts of your body?

The fat cells will not go elsewhere after the tummy tuck  procedure. The fat cells that are removed are gone forever, however the remaining fat cells can increase in size with increased weight gain. Therefore it is important to maintain a healthy lifestyle with diet and exercise. 

No, after a tummy tuck those fat cells are gone forever, but the ones that are left behind can still enlarge in response to eating more, so you will still be able to gain and lose weight as before.