You can transfer eggs easily as long as they are stuck on something that is easily moved (such as a pot or a rock). Gently lift the item containing eggs out of the water, quickly move them to the new tank and gently lower them back into the water. I have heard that any air contact will kill the eggs but I did this (with JD eggs, although I doubt that mattered) just a few weeks ago and the fry are alive and apparently strong.
Leaving the eggs in place without the parents is an option, but the mother will fan the eggs which keeps fresh water on them as well as removes the funguses eggs so the fungus does not spread to the other healthy eggs. If you remove the mother from the tank doing this will become your responsibility.
Once a JD pair is formed they are likely to spawn regularly, so unless you are extremely anxious for this brood to grow up, I would ignore the eggs and let them start over in the new tank.
Achieving a breeding pair is always exciting and learning to help care for the fry is very fun (at first)â€¦ but it doesnâ€™t take long until them spawning becomes normal behavior and finding something to do with the offspring becomes a burden. The easiest thing Iâ€™ve found to take care of fry are tank mates