Any suggestion on how to ease mild/moderate back pain on a plane/ lumbar support?

Let me try. Few suggestions, take your medication when your board the plane, take frequent walk around , if you do not use medication on regular basis, i will take some anti-inflammatory when boarding the plane and some pain medicine. If you do not want to do that you could use patch for pain relief.
Move around. The best way that i know of is to change positions about once an hour. Get up and move around if allowed. Make sure when you are in your seat that you do not slump, sit up straigt. A lumabr support is helpful.