Nice pond!
That's a lot of stones!
Obviously round stones are harder to stack than flat ones, but that's a moot point as of now.
Round stones will obviously roll out of place very easily, especially on an incline. It probably doesn't take much to get them rolling. A nudge from a foraging koi will most likely get them going.
I wonder how much more water there would be for the fish to live in if those stones were not in there?
A bare liner isn't really that bad once the biofilm covers it.
I get the desire for a natural looking bottom (and sides) and all the nooks and crannies for beneficial bacteria to colonize.
I don't know if there's much you can do at this point other than removing all the stones. I think just deal with what is happening and once in a while go in there and collect the stones that rolled to the bottom.
If you drained it and cemented all the stones, that would be a complete mess. The cement would probably be bad for your water chemistry and by emptying your pond you will lose all that beneficial bacteria.