Check realoem.com for the headlight parts for an E53 bearing in mind if you have Xenon or Halogen lights. There may be internal parts that have perished. Option 2 (you have little to look) is to heat up the light in your oven (50 to 80 C) to aid in removing the glued sections (if needed) so you can inspect what is causing movement.

I did this for a JDM Toyota many years ago with sucess (disassembly, inspection, repair and reheat to reinstall the plastic lens cover.

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2002 Euro X5 (E53) 3.0 diesel (M57)
5 speed manual with UUC short shifter.
Pre-xDrive NV125 transfer case (chain & gears).
Factory Xenon headlights, Spyder LED tail lights
Updated vortex crankcase breather.
Motorsport dead pedal, factory fire extinguisher, factory first aid kit, factory F&R mud flaps, factory PDC

Bilstein B6 shocks F&R
Redline oil has replaced "lifetime" fluids (F&R diff, manual transmission, transfer case and P/S).
Dimple magnetic drain plugs in all compartments.