No.1: What Operating System(s) are we concerned with? Depending on the OS; you should be able to find the existing files (emails) within a select arena. As others have stated, they will be '*.dbx' files. You can either move the old hard drive over to the new PC and then copy from that drive while it is a secondary D:\drive; or you could copy to a network share, or you could burn the data to a cdr in order to move onto the new PC. Make sure to "show all files"...
Look under these areas, depending on the OS...
Win98: Would be under C:\Windows\App Data\Identities\34er5t78709(example)\Outlook Express (there may be a Microsoft folder before OE folder, it's been awhile since I used Win98)
WinNT: look under-C:\WINNT\Profiles\%Username%\Application Data\Identities\{09A3C577-AFB9-4992-A600-38D752F2B402}\Microsoft\Outlook Express
Win2K & XP- very similar to WinNT in the structure - BUT, you should look under Local Settings then APP Data, not just App Data, there is a difference.
The whole object is to have the complete folder with all of the email folders transferred - so just copy the subfolder you prefer. If it's just the OE folder, then copy it and identify it appropriately and then, when you open up the new PC and start to use the OE program there, simply click on FILE\Import&Export and run the wizard by allowing it to find where that folder is now.. to import the data from! Since it's the same program, it should be very 1-0-1 in completion.
Let us know how you fare!