https://github.com/metafizzy/flickity/issues/598
Sounds there is a bit of confusion in how the previous/next arrows work for a navigation carousel. The previous/next buttons allow a user to navigate cells in that carousel. So you could swipe to the far left or far right, and it wouldn't change the main carousel. The main carousel would only change when you clicked on a navigation cell.
To enable your desired behavior, you'll have to implement your own custom buttons. See demo https://codepen.io/desandro/pen/291975a383c8bce86d733dbcb2435a07