Thank you for your question. I do not think Doxycycline or Retin-A will affect the hair transplant procedure. Please talk to your Hair Transplant Surgeon in addition to your Dermatologist. All The Best !
As you rightfully state: "it does depend on seeing your doctor since everyone is different". In general you need to wait to see the transplanted hair grow out. This takes 6 to 12 months.