Is it also possible that the dirt is from the playground?
I know I personally have to put shoes back on someone about 5 times each afternoon when the toddlers lose them while climbing, tunneling or simply toss them when they get a piece of mulch inside.
I require families to keep two complete sets of clothing (to include shoes and socks) so I can change them into clean clothing after outside time. I launder the others and put them back into the cubby. I also request kids come to daycare in play clothes, only.
Maybe that will work with your provider?