If the mulch washes down doesn't the soil also? Maybe you need a retaining wall at the low end or a thick ground cover to hold the earth in place.

The only time I have seen mulch wash from a garden is when the slope was greater then 10 percent, or there was rainfall greater than 1 inch per hour. If the mulch is being washed away so would the soil that mulch is helping to protect.

A common problem with this part of the yard is that the soil is piled up above the level of the concrete, whether it's sidewalk or curb.

I suspect much of the work of edging and weedwhacking to keep lawns back off the sidewalks would go away if soil wasn't piled so high next to them.

If the soil level was BELOW the edge of the curb there, you could mulch on top and still be level with the concrete, which would hold the mulch back.

If that's not feasible try using a border of some kind, rocks, bricks, plastic edging sunk to the desired level, etc.

Or, I've heard that a steep slope can be stabilized by sticking small twigs into the ground. Let them stick up about to the height of the mulch, so they are not visible. The mulch will catch on them and not migrate downhill as fast.