A theorem of Deligne says that compatible systems of l-adic sheaves on a smooth curve over a finite field are compatible along the boundary. I will present an extension of Deligne's theorem to schemes of finite type over the ring of integers of a local field, based on Gabber's theorem on compatible systems. This has applications to the equicharacteristic case of some classical conjectures on l-independence. I will also discuss the relationship with compatible wild ramification.