Hiya and Welcome !
Simple answer is YES !
Then it gets a little silly, but only a little silly !
First, what Operating System are you using ? i.e.Linux or Mac or Windows and what version.
This will not be complicated but it's best to know what you're running before starting with the basic ansers...
Also, when you refer to your new machine and
updating LimeWire Pro, does this mean that you had LW Pro on your new machine and have subsequently done an update of LW ? (Makes a difference... Maybe a BIG difference !)
To prepare you... Simple answer is that you simply 'copy' your old downloaded files and then load that 'copy' in to a suitable folder on your new computer's hard disk drive..
This has NOTHING to do with LimeWire. (Unless you are also talking about your downloads being your shared files in which case LW needs to be 'told' where those files are located on your hard disk.)
Easiest way to do the 'copying' is to use an external hard disk drive... If you don't have one, borrow or steal !
Other option is to 'network' the two computers. Help available if you choose to go that route.
C U
