Jennifer, I have heavy clay soil, and the trick with making paths is to put down some kind of polyester landscape fabric (not the black plastic with the holes in it), or old polyester shade cloth (or new if you don't have old, since it lasts for years and years and years), and then put wood chips or gravel over that. Rock and wood chips will sink into the clay and disappear again and again and again unless there is indestructible fabric there to keep the stuff up. Then you'll have a nice long-lasting walking surface.
Since clay only gets into real trouble when it is exposed to the sun, deep mowed grass/weeds or crushed leaves over the top or the raised bed, as organic mulch, will keep the raised bed protected from heavy rain, keep moisture in so it doesn't dry out, break down and improve the soil, and keep the heavy rain off of it. Maintain a 3" thickness of mulch if you can, no bare spots, sometimes that means adding a little more every couple of weeks.
Mulching will also maintain a moisture level that will bring the worms up into the raised bed portion, and they love the broken-down compost the mulch turns into.
In a hard downpour the clay in the raised section might flow down, but any soil would flow downward in that situation. I've found that my heavy clay soil stays really great when the rain doesn't pound on it. In my greenhouse the rain can't get to it, and it's always easy to dig and plant in. So if there is going to be some heavy rain I have a supply of sheer curtains, that I also use for keeping bugs and birds off the crops, that I put over the beds to protect it.