We will discuss differential fields in positive characteristic equipped with a differential automorphism (i.e., an automorphism that commutes with the derivation). Our main focus will be on the notion of differential-difference kernels (based on the notions of kernels by R. Cohn and B. Lando). I will try to explain when these kernels can be realised inside a differential field with a differential automorphism. As an application we will see that the model theory of "generic" differential automorphisms is tame; in particular, what model-theorists usually denote by DCF_pA does exist. This is joint work with K. Ino.