Best Practice? Disable Mail Flow prior to switch?

Good day everyone.
I am going to switch over to my new environment this weekend and would like to know if I am on the right track.
My big question is stopping the mail from flowing while both servers are active on the network. Will stopping Zimbra with "zmcontrol stop" stop mail from entering into the server?

EXPORT - OLD Server:

1. Disable the VM network.
2. "zxsuite backup doSmartScan"
3. Shut zimbra down (zmcontrol stop)
4. Shut postfix down (postfix stop)
5. Enable the network.
6. "rsync -avH /migrate/ root@:/opt/zimbra/backup/migrate/"
7. Disable network on OLD Server.
8. Shut server Down.

IMPORT - NEW Server:

1. Import Backup.
2. NETWORK: Change IP address to OLD Server IP.
3. NAMED: Change all IP ADDRESSES in /etc/named.conf and /etc/named/ files.
4. Take over SSL certificate using the following guidelines:
   --- https://wiki.zimbra.com/wiki/Transfer_SSL_certificates_between_servers
   --- https://wiki.zimbra.com/wiki/Installing_a_Comodo_SSL_Certificate_on_Zimbra_Collaboration
   ** https://ssl.comodo.com/support/csr-generation-zimbra-server.php
   ** https://ssl.comodo.com/support/certificate-installation---zimbra-server.php
5. Restart System and test server.

kind regards
aubrey