挪 implies a small, lateral movement — nudging, scooting, shifting something a short distance. It's not for big moves (that's 搬 bān). 挪用 (nuóyòng) = to misappropriate/embezzle — literally 'move the use' of money to where it shouldn't go.
Hand radical — moving, shifting, and shuffling are hand actions, so 挪 joins the 扌 family with 移 shift, 搬 move, 拖 drag. The radical also marks the very physical, 'with-the-hands' quality of 挪: not a grand relocation but a careful nudge.
Phonetic — 那 nà → 挪 nuó (initial preserved, vowel shifted). Also the phonetic for 哪 which/where, 娜 graceful, 钠 sodium. 那 here lends only its sound; the 'that-ness' meaning of 那 has no semantic role in 挪.
