Is nausea and neck pain common with Dysport in the chin area? If so, how long till it goes away?

This reaction is not related to Dysport. If you believe there's some association speak with your injector and discuss your concerns.Warmest regards, Dr. ALDO

Patients who are undergoing injections with neuromodulators to reduce muscle activity should be well educated as to the indications for treatment, the actions of the medications and the expected timing to see results. Talk with you plastic surgeon about the potential side effects and have a plan for what to do if the results are not what you had hoped for. Remember, this is a medication which limits muscle activity, it is not liquid magic. 

No it is not common to have internal symptoms from local injection of dysport.  It is likely coincidental.  Please see your physician.  Best, Dr. Emer.