I only carry an extra outfit for the baby, and I have 5 children ages 10 and under, so never past 2.5. A messy outfit from eating isn't the end of the world. Messy diaper blow-outs (or potty training accidents) are really the only reason I'd want to have a change of clothes with me.
If you're staying off-site and driving in, you can keep an extra outfit in the car.