I use 3 computers: an older iMac (LR5) and MacBook (LR3) in one location, and a new iMac in another. I installed LR 5 in the new iMac, thinking I would then upgrade the MacBook and have LR5 in all three computers, but found that Adobe only lets you put it in two. So now I want to move LR5 from the new iMac to the MacBook, and put LR6 in the new iMac (because I've found the LR5 sliders etc are *sluggish* in El Capitan). Is this move possible?
Yes, you can uninstall the Lightroom 5 from New iMac and install it to the Macbook.
You can buy Lightroom 6 and install it on the new iMac Machine.
Adobe allows for 2 installations in parallel but if you read the license agreement, you will see, that you cannot use both at the same time. For a single person this is an academic problem, but if you work as a group, you would need a license for each user.
Now regarding your question: I doubt that the catalog files are interchangeable between LR5 and LR6. I would need to install my LR5 again to check that. But as a matter of a fact, I would not like to use 2 different versions of a program.
May be it is OK for you to deregister one machine, before starting working on the other one. Again, here I'm a little bit spoiled, but I would hate working an an old and slow machine, when having newer ones available.