Neostability transfers in derivation-like theories

Motivated by structural properties of differential field extensions, we introduce the notion of a theory $T$ being derivation-like with respect to another model-complete theory $T_0$. We prove that when $T$ admits a model-companion $T_+$, then several model-theoretic properties transfer from $T_0$ to $T_+$. These properties include completeness, quantifier-elimination, stability, simplicity, and NSOP$_1$. We also observe that, aside from the theory of differential fields, examples of derivation-like theories are plentiful.