Q.Why My Kaffir Lime Dropping Leaves? Why Does The Leaf Has Browning Edges?
Lime plant brought indoor November 1st. Water once a week, soil kept moist using moisture meter. When brought indoor it has full foliage. Some leaves turned brown from the edge inward and started dropping the top part of the leaf, then the second part of the leaf dropped a few days later. At the same time, new sprouts came out but the new sprouts started to brown and the new leaves are very thin and light yellowish green. The older leaves all dropped of. Now, the new leaves from the new sprouts are starting to drop too. The bare branches are browning and dried out from the top. The lime is indoor with 10 to 12 hrs of grow light full spectrum. There is a couple new budding shoots at the root but it turned black.

Certified GKH Gardening Expert
Plants have a tough time acclimating to the indoors after spending time outdoors in bright light and higher humidity. The Kaffir lime needs full sun, so, even though you use a grow light, it should be close to a sunny window. Also, it probably needs more humidity. You can mist it or set it on a pebble tray. These lime trees do like moist soil, but when indoors, it is easy to overwater. Let it begin to dry between waterings. It may not need water every week.
The black budding shoots is concerning. Root rot may have set in. I would repot the plant in fresh soil and a clean container with drainage holes. These articles should help:
https://www.gardeningknowhow.com/edible/fruits/lime/care-of-your-kaffir-lime-tree.htm
https://www.gardeningknowhow.com/houseplants/hpgen/what-is-a-pebble-tray.htm