Yes the script will be honored until all your refills are used.
This is a duplicate post... I will/have responded to your other post as you added a bit more explanation.
~Tuck
Should be ok, all they need is the script for their records, I had refills on a script and the doctor had closed and moved to another city and I had no issues. Good luck! If its a refills call it in, if there's an issue they'll call you. Don't volunteer any info, they don't know you got a call.