This is always a tricky one. I have seen people rake their runs with a leaf rake daily to remove droppings - I do the same in my grass field runs every couple of weeks when the droppings are dry but the only other thing you could do is remove the top couple of inches and replace it with new soil, or wood chippings / bark / shingle / rubber chippings - people seem to use different coverings that they can rinse through and then change eventually when they get too mucky. I have even seen turf being used - but I would be careful with that as there can be chemicals used on the turf.
I am in a similar situation. I have one run in my garden and I have used 20mm smooth stones although they have got to the point where no amount of washing is going to help. I plan to remove the stones this summer and pave the whole run so I can wash it. The chooks have daily access to a field and grass and are rarely kept in (only during winter when it is really wet in the field) so this should work for me.
Tim