The Go-live which is the final step requires a change in the DNS entries. While the changes are done by changing a few settings, it may take time for the new values to reflect across all the DNS servers around the world.
During this transition period, some mail will be delivered to the old email servers even after the changes to the DNS are done. So to avoid mail being split, it is advised to change the DNS settings in non-peak hours. Usually around 8 pm for a company working 9-5.