In much abbreviated form...
1. Take the files you want to edit else where and select them all on your PC.
2. Export as a Catalog including the Image Files.
3. Transfer that exported catalog to the Mac (External Hard Drive is a common method)
4. Open the transferred catalog on the Mac and work on your images.
5. When done, select all and Export as a Catalog.
6. Transfer this new exported catalog with the edits back to the PC.
7. Open your Original Catalog on the PC and Import From Another Catalog and point to the Catalog from Step 6.
Thank you for your reply. If I can ask you more specifically, I will be shooting in a remote location and then want to import files to a mac, work on some of the images in LRE and then when I leave take them with me. So if i understand I can do this easily my exporting the catalog of files I worked on on the mac and then importing that catalog into the catolog on my pc when I get home. Could please confirm this? Thanks very much.
Yes. In this specific case you are replacing steps 1-4 with "Import onto Mac and Work on your images"