How to transition to Google Apps without losing e-mail

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http://serverfault.com – We're about to move a client from a "regular" Courier Mail Server at another provider's server to Google Apps with us. So at the same time, we're switching their DNS records to "point" to our web server and Google's MX records. How can we avoid e-mail getting lost in the transition? Should we shut down Courier beforehand so that it doesn't receive any e-mail until the DNS changes have propagated? What will happen to e-mail that is sent to Courier but not received? Also, what about users who don't check their e-mail very often, how do we move their e-mail from Courier to Google Apps when they (HowTos)