A default choice for this kind of task is the olden nohup command; prefix
nohup in front of your command will make it continue even after you closed the terminal:
nohup rsync -azvv -e ssh root@A:/home/some_path /home/target_local_path/
If you close the window before rsync finished, the rest of its output will be written to a file named
nohup.out in the directory you started rsync.
However, in case that after you starting rsync, you also wish to do other things before logging out, append
& at the end of command line to make it run in the background. You would need to redirect its output to file too in this case, since rsync would emit output that would interfere with later commands:
nohup rsync -azvv -e ssh root@A:/home/some_path /home/target_local_path/ > rsync.log 2>&1 &