Answer: To some extent this depends on the size of the cobbles or stones. If they are fairly large, they should stay in place. If they are small, then the freeze/thaw action of the soil in spring and fall will cause it to move up and down and become mixed into the soil. Using the fabric might help somewhat to keep them separate, but in my experience smaller stones still move and the fabric just scrunches beneath them. If there is fabric below then it prevents soil to mulch contact and thus drastically slows the decay process of the mulch... (Organic mulch is much healthier for plants since it does break down over time and helps feed the soil.)
If the stones are on soil without fabric, the mulch will rot down over time and work its way down below the stones. If you keep the area swept or blown clean of incoming debris (such as autumn leaves and spilled mulch) the current mulch over stone issue should go away on its own if the stones are large enough not to be frost heaved.
The best way to avoid all that is to lay a gravel and sand base, compact it, and set the cobbles on top of that just as you would do a semi-permeable patio installation. I hope this helps you evaluate your options.
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