My ISP used to use some of the Unix utilities (rsync, etc.) which keep each server exactly the same....until the very last moment.....
Another way to make certain is to, on the original server, just shut it down for NEW posts during the final DNS move. As mentioned above, it takes a couple minutes these days...godaddy seems especially quick! Last time it happened so fast it flustered my ISP - like less than 3 minutes!