You can find many ways of cloning old server to new hardware, create diskimage and move to new hardware and if you have identical hardware use
dd or similar tool to copy disks block by block, but if you have new hardware and want to move server without downtime and with no configuration change there is nearly no solution.
I used vestacp, ejabberd in my servers so i had to make changes even after moving some files, as IP changes every where but it is still far better solution than installing and configuring each package manually.
We used rsync to copy whole server to new server except following files
because on new network you will never wish to propagate old network configuration,device specific configurations, installation specific configs.
Create a file with following directories and then execute
rsync will Synchronize all your files from remote to local.