It's not going to be cost-efficient to move to the moon unless and until there is commercial demand for scrap material on the moon and equipment to process it. A lot of delta-v is needed to transport stuff to the moon. On the other hand stuff in LEO naturally deorbits with a certain timeframe and can be accelerated with a small nudge, a dragsail or possibly even laser ablation, and it's really not very far to go if you decide to actively deorbit it.

You'll likely get recycling in orbit (where the spacecraft are) before the moon (which has abundant aluminium anyway) first, so the compromise would be shifting debris in LEO to storage orbits with longer decay times