Here's what should work (on CentOS platforms only) just using tar. The idea is to backup everything except /dev, /proc, and /tmp. On hosted VPS platforms, I think you would also want to remove /sys and /boot since your new VPS will already have one that is needed on the new platform.
Before you begin the backup, you need to gracefully shut down asterisk, mysql, and apache.
After the backup is complete, you need to remove /var/lib/dhclient, /etc/udev/rules.d, and /etc/sysconfig/network-scripts directories as well as /etc/fstab from the backup, or you cannot restore it to a new server that will boot and have network connectivity. Stated another way, the existing disk and network setup in your new container needs to be preserved when you restore your backup.
We also recommend creating your replacement VPS in as close to identical design as the previous VPS, just as if you were starting over, i.e. if you're using Incredible PBX 13-13 with CentOS 6, build the new system the same way NOT with CentOS 7 or Ubuntu or Asterisk 15.
Be sure to restart Apache, then MySQL, and then Asterisk in that order. And copy /tmp/backup.tar.gz to a safe place far away from your VPS.
Our 2.5GB backup on Vultr took about 17 minutes to build using the script below. Then we copied it with scp to a VPS of another provider in a different city in about 15 minutes.
Code:
amportal stop
service mysqld stop
service httpd stop
service sendmail stop
cd /
tar -cf /tmp/backup.tar /bin /etc /home /lib /lib64 /media /mnt /opt /root /sbin /usr /var
service sendmail start
service httpd start
service mysqld start
amportal start
tar --delete -f /tmp/backup.tar etc/udev/rules.d
tar --delete -f /tmp/backup.tar etc/sysconfig/network-scripts
tar --delete -f /tmp/backup.tar var/lib/dhclient
tar --delete -f /tmp/backup.tar etc/fstab
tar --delete -f /tmp/backup.tar etc/resolv.conf
tar --delete -f /tmp/backup.tar etc/hosts
tar --delete -f /tmp/backup.tar etc/hostname
gzip /tmp/backup.tar