Separating DAGs using separators consisting of lists of nodes and all ancestors
In this answer i assume that $u$ is an ancestor of $v$ if $u$ can reach $v$ by a directed path.
This is basically as hard as Set Cover (Given family $F$ over a universe $U$, find smallest subfamily $F’$ of $F$ whose union is $U$). To reduce from Set Cover:
Make a vertex for every set in $F$ and for every element in $U$. Make an arc from every element to every set that contains it.
Now picking a list corresponding to a set means deleting the set and all the elements in it.
For every element add some new dummy vertices and add arcs from each element to its own private dummy vertices.
Now observe that as long as all elements have not been deleted there are still components of more than $O(1)$ vertices left - an element plus its dummies.