Hey there, I have been growing flowering vines for about 10 years and have recently started an online business selling them.I have relocated them. You will have to dig them up carefully. Dig about ten inches around (about the size of a hanging basket pot). Morning glories usually do not root too far down and I have found most are will do fine if transplanted, moonflowers at time can be picky but if they are that large tou should be fine. Pot them for the trip.

Keep them well watered in advance , during and after any transplanting...use a large clear leaf/recycling bag (as a temporary humidity chamber) to cover the plants if they wilt too much...

Maintaining root integrity and maintaining hydration is the most important and gradually re-introducing them into full sunlight...

Do NOT use any type of fertilizer on the plants although using the the products designed to help lessen transplant shock may be helpful if not overdone...

regards,

Ron

P.S. - The reason I mentioned clear bags and / or any type of plastic covering is because the clear / transparency allows any available light to reach the plant in addition to allowing you to (see through any plastic and) better monitor the plants condition.

Clear plastic is another tool which can (act as a partial sun-shield and) enable the transition to full unfiltered sunlight.