If yer bought it new or used from a LR main dealer, had it serviced by them regular, had several LR's over the years... all or some of these options... will give yer a betterer chance of LR assisting with the whole or some of the cost of repair, even when out of warranty, if they see evidence of your continuing custom. Repair would have to be done at a main dealer. If yer haven't done any of the above then yer chances are reduced. As said it's worth a go.
Yer FL2 has drive away locking, which locks the doors when yer drive away. It can be enabled/disables in the vehicle set-up menu in the dash using the buttons on the rhs of the steering wheel. I'm wondering if for some reason it's trying to lock itself as it thinks it's not locked, when driving. That doesn't account for bumps setting it oft but yer never know. When yer drive away and it locks, press the unlock button below the radio screen (to unlock it) to see if this stops the random locking. Pressing the unlock button under these circumstances will stop it locking again while yer moving. Or yer could try disabling drive away locking in the dash menu to see if that make a difference. None of this guarantee's a solution, but it may help to break down the problem.