Prevent Dirt Movement Down Slope Through Gap Under Fence
To further isherwood's comment on your original post, once your neighbor's dirt is piled high enough on his side of the fence, the dirt and water will start to come through the gap between the bottom 2 boards instead of under the bottom board. Then you're back to the same problem, just higher. Until the bottom board rots out.

Harper's comment about working with the neighbor to reroute the drainage on his side and lowering your fence to restore his view is probably the best suggestion.

If the neighbor is unwilling to work with you, then your best bet is probably going to involve a fair bit of expense on your side - installing a 2nd French drain on the other side of the walkway to capture his runoff before it hits your walkway. 

Getting the water off on the uphill side will eliminate the flooded walkway, leaving just the dirt behind (if there's fine enough filtration on top). This will, unfortunately, require some regular maintenance on your part to keep your uphill drain clean. It's one of those trade-offs in life - clean the drain or walk through the puddles.

After getting some estimates to tear up half your walkway, install the drain, and reinstall the walkway, you may want to go back to the neighbor with some cash in hand and offer to help pay for the drainage work on his side instead. Note: I'm not suggesting you pay for all of it, or even the majority of it, just some as a goodwill gesture and because it will be cheaper (and less disruptive to you) than doing the work on your side. A cash contribution toward the work and an offer to lower some of the fence to restore at least some of the view without giving up too much of your privacy may go a long way toward getting this solved properly. If you go this route and the neighbor is agreeable, I'd suggest putting it all in writing before starting, just to be safe. I don't think a lawyer would be necessary - generally putting it on paper with both your signatures (each of you retain a copy) should be sufficient. That's a question for law.se, though, not here.