I take the supplement NAC which also removes toxins from the body. Will it make my Dysport/Botox stop working sooner?

I do not believe your supplements will affect the time Botox/Dysport will last. Both typically last about 3 to 4 months!
Best. Dr. Aldo

Taking the supplement NAC should not affect the results of a Botox or Dysport treatment or shorten the longevity of the result.  

Thank you for your question. No, taking the supplement NAC should not affect the results of your Botox or Dysport treatment or shorten the longevity of the product. It is always best to address any concerns you may have with your injector. Best of Luck!

The supplement should not interfere with your Botox/Dysport results. It should not shorten the longevity of the results. 

Supplements of any kind should not affect the action of duration of neuromodulators.Dr Crabtree

Botox works by disabling the nerve signal to the injected muscles. Clinical studies show that the actual Botox substance is absorbed within 4 hours after injection. Botox lasts 3 to 4 months because that is how long it takes for the nerves to regrow their ability to send signals to the muscle, not because the Botox is physically in there for that long. With this in mind, it is unlikely your supplement will have an impact on your Botox results.  Because there are minimal studies on supplements in general, there is no study I can point to that will give you a definitive answer. If you want to be on the safe side, just avoid the supplement for a few days before and after your treatment.