I was getting faults on an 06 LR3 HSE, but somewhat opposite where it would slip into neutral at rolling stops. However did loose Sport and Manual modes at times. Transmission faults sometimes. I decided to do the Transmission filter update kit, thought it made a difference, but it came back.
I discovered a significant symptom of the 'D' on the dash display at bottom of speedometer.. The 'D' would go out whenever the problem came up. I would put more pressure on the Shifter, to get it deeper into Drive, and D on dash would light up and I could drive again. That seemed mechanical at first.
A word to the wise... If you mess with the linkage lever, block your wheels, set parking brake, etc.
Transmission issues get you under the car a lot. One time I was on a very slight slope, just outside the garage, messing with adjustments. LR3 was in Park. After loosening the bolt on linkage lever, it doesn't take much pressure to move the lever between - Park, Reverse, Neutral and Drive - or loose track of which position it's in. Well it moved, suddenly ending in neutral position, I'm laying on my back and witness an optical confusion as the bottom of the LR3 is moving. The big heavy LR3 and gravity met. Luckily, I was able to scoot myself out and block a wheel. No serious injury, just a little road rash on a kneecap, a few prayers and some gin.
That delayed me working on it for a few days. Getting my courage back, I adjusted the linkage lever on it's cable to the console shifter, It's located on passenger side of transmission. This fixed the situation, but it would return. There's a rubber bushing inside the linkage connector that gums up with road crud from miles. It's supposed to rotate to allow cable to extend into the Park and Drive positions at opposite ends of it's travel. Anyway, I loosed the connector bolt and and then rotated that bushing and applied some lithium grease, to get it moving better. That also seemed to help for awhile.
Next trip under car to adjust linkage, I noticed an electronic connection behind the linkage lever, that appeared like it was not seated properly. It likely got unseated doing the initial transmission kit and all it's associated knuckle busting, but that's hard to pin down. In any case, I re-seated it, and no more transmission faults or dropping out of gear.
On the the next LR3 puzzle, a high pitched, and highly irritating whining noise under the hood, that's came now that I'm driving it again.