06 LR3 keeps moving out of drive and into neutral

Usually this is caused by the shift linkage at the transmission binding, which will cause tension, and even though drive is engaged, as soon as you get a bit of a torque shift, it'll drop into neutral.

The linkage can sometimes just be removed, cleaned, and lubed up, but often requires replacement as the plastic will get damaged during removal if it's real bad.

I second Roverguy7.
This is a typical example of gear linkage slipping.I cleaned and lubricated mine proactively long time a go and it is still fine, but replacement is fairly easy as well.

I just had that issue few weeks ago, cleaned and oiled the linkage (i.e. sprayed WD40 and silicon lub, lol) and all is fine now, I even feel shifting the gear from P to D is much smoother.
But the real fix is described here:
DISCO3.CO.UK - View topic - Autobox Gear Lever pivot - How To

Would this be the same fix on an LR3 2008 Land Rover?

Yes