In the old computer:
Menu >> File >> Export >> Export contacts as .vcf files.
Click Next. If you have more than one account, specify which one you would like to export from.
Then select if you prefer all contacts exported in one .vcf file or if each contact should be exported into a separate file. Both methods or fine.
In the new, once eM Client is installed.
Menu >> File >> Import >> Files >> Contacts as vCard (.vcf)