Q: Can I add fall leaves to the run?
A: Yes, but you probably won't want to add too much, too often.
There may be some entertainment value in adding a few fallen leaves sparingly to your run, because your flock will enjoy scratching through them to look for bugs. If you scatter some treats through them as well, their excitement may also entertain you! However, chickens won't eat fallen leaves, so they aren't something to forage on directly. And in too much quantity, fallen leaves in your run can cause problems.
Why? Well, for one thing, they don't make good bedding, since they're not absorbent. In many cases, they will get wet from the droppings, and then rot or mold in the run. That's not so fun (or healthy) for your chickens.
And if you previously had grass growing in your run, a big pile of leaves could cause soil disturbances and kill what the flock has been enjoying. Walnut leaves, for instance, contain juglone, a natural herbicide. (Walnut trees want to make it difficult for non-walnut trees to compete too close beneath their canopies.) Maple leaves have a pH of 4.3--too acidic for many plants. And a big impermeable pile of leaves of nearly any sort can simply smother what's beneath them.
If you have lots of fallen leaves in autumn you're looking to find a good use for, consider adding some mulched leaves to your compost pile! (Even then, some leaves like Walnut are not going to be good for composting, either.) When you also have chickens, you'll have chicken manure and bedding such as pine shavings to mix into your compost pile, too. The fully transformed compost is great when mixed into your garden or landscape beds.
If you're looking for the best use for your fallen leaves, you probably can't do much better than learning about efficient composting and using them in your pile along with your chicken manure. Tomatoes love chicken-manure compost!