I was surprised by this because I thought rsync was smarter than that. When I tried it locally, the transferred files had a new date created - so the files were newer on the synced folder and hence "different".
You can add a -u flag to the rsync command which ignores the new timestamp on the destination. if you treat the source as your master, that should work,
rsync -uv ~/source/*.* ~/Documents/Synced/
sent 1,394,371,905 bytes received 1,004 bytes 13,092,703.37 bytes/sec
total size is 1,394,027,448 speedup is 1.00
total size is 1,394,027,448 speedup is 1.00
And the second time:
sent 2,089 bytes received 12 bytes 4,202.00 bytes/sectotal size is 1,394,027,448 speedup is 663,506.64
You can also skip some of the Moodledata folders - there is a current discussion on backup here.
Of course you must still back up and restore your database to the new server - rsync could handle that too. Might be quicker that a restore if you use batch files and cron.
{Edit} From the man page -t preserves timestamps for smarter copying.
rsync -tv ~/source/*.* ~/Documents/Synced/
Much better!