When a mailbox is moved from Exchange 2003 to Exchange 2010 it is taken offline. If the user is using Outlook in cached mode, the emails they currently have in Outlook will be visible to them but any emails they receive while their mailbox is in transit are held and placed in the Exchange 2010 mailbox once the move has completed. If the user uses OWA, they will not be able to access their mailbox. I'm not sure how Outlook online mode would react, but my (possibly fuzzy) logic says it should act the same as OWA and not work very well.
I'm not entirely sure suspending the move will allow mailbox access, and I'd probably say try it on a test mailbox and see what happens. The worst that will happen is you'll get an error message.