Does Botox injected in the face travel to a breastfed baby?

Botox is an intramuscular treatment that doesn’t travel through your bloodstream. That said, I never advise patients to get unnecessary cosmetic treatments while pregnant or breastfeeding. You just don’t want to take the risk. Plus, doing so creates the kind of worry that you’re now undergoing. 
Take your baby to the pediatrician to find out what’s wrong. Don’t get a touch-up to your Botox if you’re still breastfeeding at that time (also, don’t worry anymore about the Botox you already have). And please consult a board-certified dermatologist who is an expert in Botox injections for your next treatment; I always ask my female patients if they are pregnant or breastfeeding before administering injectables.

While there aren't studies about Botox and breastfeeding, botox will not travel. It will take 1-2 weeks for final results with Botox. Best, Dr. Emer

I do not recommend or treat patients with Botox or Dysport while they are pregnant or breastfeeding. I suspect your baby will be unharmed but I recommend avoiding injectable cosmetic treatments until you are finished breastfeeding. 

Botox does not travel throughout the body so you do not have to worry about any effect on your baby

Hello!  Thanks for the question.
I personally set pregnant women and breastfeeding women as Off limits. I do not inject them no matter what.   It is an elective procedure and not worth any risk that it may cause, and especially the liability that could be blamed on the injection.  
Having said that.... some providers will inject and if done correctly it is probably safe.   If your injector knew you were breastfeeding, they may have used a smaller dose or diluted dose and would have been extra cautious I’m sure.  
 The injection is into muscle or just under the skin, near the muscle.  It SHOULD NOT get into a vessel and travel through the body and enter the breast milk.  But it COULD be incorrectly injected into a vessel.  A small amount into your system may not affect you,  but could effect a baby. 
There are no studies to prove whether botox while breastfeeding is safe or not.   If a baby did get a large dose of the toxin, they would have signs of botulism.   This includes, weakness, “floppy baby” with no muscle tone, trouble breathing, loss of bladder control, droopy eyelids, nausea, stomach pain and diarrhea.    
Your child may just have a stomach bug, but if you aren’t sure, make an apt with your pediatrician just to be sure.   Good luck with everything.  Hope your child is feeling better.