I moved a small Russian sage to a new location besides AC unit in the morning.
By the afternoon, all the leaves are dried and curled up.

Is Russina sage not adapting to move?
Or did the air from AC unit kill it?

You tell me? Did you have to dig up weeds or 4 inches of mulch that kept weeds down, or did nothing grow there before?

Russian sage likes cooler air, sometimes plants go into shock when you move them, wilt for a week and come back, and sometimes, they cannot handle the move- they die. Many things make a difference- too hot, didn't take enough roots, plant too big, you didn't give us pictures so it is hard to tell.

Perovskias (neither Russian nor sage btw ) have wide-reaching root systems that are hard to completely dig up, so are easy to break. Also, moving them once they've leafed out in spring is touchier than doing it while they are dormant. However, I would have expected the leaves and stalks to go limp and droop, not dry up. Was it a hot day? did you water in and protect the plant from the sun?

If it's a central AC unit you set it by.........not a good idea. When the unit is running it blows really hot air around the unit. At least all of mine at various houses have done so. The hot, dry air fanning the leaves when it kicks on would cause them to curl and dry up and ultimately kill the plant if left there too long. If you planted in-ground, I'd suggest digging it up as carefully as possible & move it (but that might kill it off from transplant shock). If in a pot, move it to a cooler place immediately, preferably shady place, water it well and hope for the best. Be prepared that it just may not survive.