We study generalised automata (in the sense of Adámek and Trnková) in Joyal’s category of (set-valued) combinatorial species, and as an important preliminary step, we study coalgebras for its derivative endofunctor $\partial$ and for the ‘Euler homogeneity operator’ $L\circ \partial$ arising from the adjunction $L\dashv \partial \dashv R$. The theory is connected with, and in fact provides relatively nontrivial examples of, differential 2-rigs, a notion recently introduced by the author putting combinatorial species on the same relation a generic (differential) semiring $(R,d)$ has with the (differential) semiring $\mathbb{N}[\![ X]\!]$ of power series with natural coefficients. The desire to study categories of ‘state machines’ valued in an ambient monoidal category $(\mathcal{K},\otimes )$ gives a pretext to further develop the abstract theory of differential 2-rigs, proving lifting theorems of a differential 2-rig structure from $(\mathcal{R},\partial )$ to the category of $\partial$-algebras on objects of $\mathcal{R}$ and to categories of Mealy automata valued in $(\mathcal{R},\otimes )$, as well as various constructions inspired by differential algebra such as jet spaces and modules of differential operators. These theorems adapt to various ‘species-like’ categories such as coloured species, $k$-vector species (both used in operad theory), linear species (introduced by Leroux to study combinatorial differential equations), Möbius species and others.