Export to EndNote Web and then updating the desktop library will not work. The Web and desktop versions do not sync. You would need to create a new library each time and can cause the records in the desktop versions will have different record numbers on different machines, depending on how the library is sorted, or if you made any deletions.
I recommend you either use the Web version without the desktop version, or transfer the library back and forth between computers.
The Web version can do CWYW with a plug in. Then you can log in and work off the same library from anywhere (assuming you have Internet access). The draw back is if you need to modify an Output Style. This is not possible on the Web version right now.
Alternately, you can move the library from computer to computer on a jump drive or via email. To do this you either need to create a compressed library (it is an option under the File menu on newer versions of EndNote and is the safer option) or copy both the .enl file and the .DATA folder with the same name to the jump drive (or zip both to email) and transfer both files to the other computer. Each time to move computers you would need to transfer the library if you made changes.
I sync the folder between computers. I use a service called Sugarsync to do it (it also keeps a backup of my files in the cloud so I kill two birds with one stone). There are free services that do the sync only.