Hi DesiDysport and Botox weaken muscle. The little lines you are referring to are due to muscle activity and also from loss of fat volume. However, BOTOX and Dysport can cause unwanted muscle weakness which is a very bad thing around the mouth--think drooling soup. So the doses are very limited which means the effect is quite transient.A much better approach is using a filler like Restylane to place a bead of filler along the vermillion edge of the lip. This helps iron out the lines, helps the horizontal width of the lips and avoids the trout lip look we all know too well.
I have tried BOTOX for vertical lines that surround the lips, and have found that Injectable Fillers are more predictable and work better.I've attached a link demonstrating this technique.I hope this helps, and best regards.