Currently, I am assisting with the technical planning for an email service provider migration for my employer. One of the things to take into consideration for this type of project is reducing the impact of any possible down time with email.
One of the ways to reduce any impact is to perform such a project over a weekend (when normal business is not taking place). Additionally, you may considering changing the default time-to-live (TTL) for the MX records. In our case, the default TTL is set to 86400 (1 day) so we are changing it to 7200 (2 hours). This will make sure that when we change the MX records for our domain all of the DNS servers around the globe will receive that change within (2) hours. Therefore, theoretically, any new incoming email while be forwarded to the correct mail servers within (2) hours of the MX record changes.
As always, with any changes made to your existing infrastructure always plan carefully and document the changes (i.e. change management or change tracking).