Sorry to say this but that sounds exactly like water in transmission (or coolant).

Mine did this, at a stop felt like driveline tension was being released. Brake to stop, then it would downshift (normally), wait a second then give you this nudge/kick/soft clunk. Felt like something was binding up and releasing after every stop.

Immediately get your trans checked, have it flushed, change fluid, anything. Unless you do a LOT of deep water crossings (trans breather is right below windshield), it could be a coolant leak or something unrelated. I hope it is not water but I had water in mine and it did just that.

As for DIY, get a buttload of fluid, get the metal pan kit (do a google) with removeable filter, take the heat shields off, drain tranny, undo pan, snip the straw/tube on the pan/filter, and install the metal pan, then filter. Refill and level. Guess it needs to be idling. I haven't done this job so can't comment on specifics. Yes its expensive for the change, I had a pan changed with new fluid at $1200 ish.

 

I just changed my transmission fluid, twice. Flushed it first, then swapped to the metal pan. No evidence of water in the transmission fluid that I could see.

I reset the transmission adaptations and went through the procedure involving lots of coasting to a stop and it seems to have made a difference.

 

I'm glad you sorted out your problem. I'll be swapping my tranny pan, mechatronic sleeve and filter soon to the metal sump. I've had the kit for about 2 years now, just need to motivate myself. Did you have to contact Gap to get the tranny software update for your LR3?

 

That was the next step if the transmission adaptation reset didn't improve things.