Just bought some of these for this same problem. Installing them this weekend. Just want to use it as a dam to keep the mulch in where it is supposed to be.

https://www.homedepot.com/p/302999894

 

My property in general and several of the landscape beds have significant slopes. I have brick edging as well as plastic edging in some of the steeper beds. For many years, Scott's Earth Grow was my go to mulch. Even with the double edging, after every rain storm I would have to rake up the mulch that floated over the edging. This year I got tired of hauling and installing 40 plus bags and paid a landscaper to install double ground hardwood mulch which is much finer than the Scott's. Best decision I have made in a while. No strain on my back and once watered in, the finer mulch sticks like glue. It has rained most of the past week, all beds are intact.

 

Got my edging installed today. I'm happy with the way it looks. Blends in from far away.

 

I know it may seem a bit counterintuitive, but have you tried adding more mulch?! Unless you have an overflow of water from gutters during heavy storms, properly mulched areas should not be washing out I would think.

 

I just use the black metal edging stuff between my beds & driveway. It's cheap, easy to install, does the job well enough and is thin enough to not be an eyesore.

If the beds have any slope to them, make sure you're using hardwood mulch and not pine bark. I made that mistake once. The hardwood stays put pretty well, but the pine bark is lightweight stuff that's easily carried by water or wind.

 

You can use plastic paver restraint to hold the mulch in place. Works and looks great. You can get the flexible one so it would go around your curved walkway and still stay right against the concrete.

 

Pine straw has worked better in my beds. It does wash that much at all. Once it settles it kinds mats together.

 

Can you direct the downspout splash block to direct the water more onto the driveway?

If not, you could use Mulch-lock adhesive from Home Depot, I've used it on some pretty steep grades and it works great. Since then went to pine straw as it stays put much better on its own.

 

I installed "no dig edging" and it's cheap, super easy to install, works well to keep mulch in the bed, and you can barely notice it from the street:





You can find it at Home Depot & Amazon and you can get the straight top if you prefer: https://www.amazon.com/dp/B00HF4VLWK/ref=cm_sw_r_cp_apa_i_SdqcFbY4K4608