In this paper we consider the following three coloring concepts for digraphs. First of all, the generalized coloring concept, in which the same colored vertices of a digraph induce a subdigraph that satisfies a given digraph property. Second, the concept of variable degeneracy, introduced for graphs by Borodin, Kostochka and Toft in 2000; this allows to give a common generalization of the point partition number and the list dichromatic number. Finally, the DP-coloring concept as introduced for graphs by Dvořák and Postle in 2018, in which a list assignment of a graph is replaced by a cover. Combining these three coloring concepts leads to generalizations of several classical coloring results for graphs and digraphs, including the theorems of Brooks, of Gallai, of Erdős, Rubin, and Taylor, and of Bernshteyn, Kostochka, and Pron for graphs, and the corresponding theorems for digraphs due to Harutyunyan and Mohar. Our main result combines the DP-coloring and variable degeneracy concepts for digraphs.