Can previous emails be transferred to a new iPhone?

We recently bought a new iPhone 5S to replace an older iPhone 5 for one of our employees.  He has a lot of business emails that he keeps on his phone for reference.  Is there any way to transfer those old emails to the new phone?  He is getting new business email on the new phone, but he does not have access to the older emails.  They are on his laptop, but he would like them on his phone, too, if that is possible.
Any help or suggestions would be greatly appreciated.
David Atchley

Unfortunately, the salesperson that was here at the office was not the brightest bulb in the pack, so I'm not sure what he had or had not done when I got downstairs.  His contacts and apps synched, but the mail did not.  I'm going to check his sync settings and see if that helps.  (Per Axeman).
I'll keep posting results, if I get any.  LOL
David Atchley

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