Drain won’t work that way. The pump that’s part of your washer probably can’t push it that far or at that height. Might try running down then up to your sink. Make sure you maintain an air gap. If you want to do it your way you will probably need some sort of booster pump.
Looking at your pics, it looks there are underground valves right there. Check if one of those is leaking and the water is causing the soil to migrate away.
If they are moving vertically it sounds like the ground under them is subsiding. Either that or it’s being removed. Is the whole fence doing this or just one or two spots.
What type of post and what is it set in? Are they wood in dirt with no concrete? Are you saying that it is moving down vertically? Something hast make it move, or displace what’s under it. Maybe remove them one at a time and set them in concrete. Hard to help more without more info or pics.