If you run public images, and mount a host volume, it would to be simply copy the files to the new host, and start it the same way as you did on ServerA.
Can you tell us a bit more about your setup?
Maybe share your docker-compose file (if you use that)
@terpz is right if you share how do you set up would be easier… but in all cases I would suggest you to use RSYNC instead of CP on bash… which will sync your servers, or your backup!