Can you keep it in covered bins, with a tarp or other water impermeable cover? If you have small quantities, you could compost in a garbage can with drainage holes. That way you just throw the top on when bad weather hits. I sometimes wondered about using an old kiddie swimming pool with drainage holes and a tarp. But, I'm fortunate enough to live where flooding is not a common problem. Maybe others will have other suggestions.

martha

Saw dust mulch might stay in heavy rain. Don't know for sure though.

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Grass clippings seem to stay in place. I don't have much of slope but I do get heavy winter rains.

to sense
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Mulches placed on even slightly sloping sites can be washed away during heavy rains and the only way to prevent that is with a barrier of some kind. There are erosion control fabrics, tubes, etc. available or you could use landscape fabric placed vertically or burlap either in a long sheet or rolled into a tube that is filled with straw, compost, or something similar.
Do a search for erosion control fo more ideas.

I saw a low-tech technique once for erosion control on steep slopes that might work for mulch. This guy was taking twigs and sticks and sticking them into the ground vertically, sticking up only an inch or two. Anything laid down would catch on them and stay in place. He was laying more sticks on top of the ground that would interlock with the little stakes, but it would probably help with mulch too. I have not tried this for mulch but it might work.

Try to get mulch that has long fibers rather than chunky types like bark. They will interlock better.