Lexicographically minimal topological sort of a labeled DAG
With multiple copies of the same label allowed, the problem is NP-hard, via a reduction from cliques in graphs. Given a graph $G$ in which you want to find a $k$-clique, make a DAG with a source vertex for each vertex of $G$, a sink vertex for each edge of $G$, and a directed edge $xy$ whenever $x$ is a vertex of $G$ that forms an endpoint of edge $y$. Give the vertices of $G$ the label value $1$ and the edges of $G$ the label value $0$.

Then, there is a $k$-clique in $G$ if and only if the lexicographically first topological order forms a sequence of $k$ $1$'s and $\tbinom{k}{2}$ $0$'s, with $i-1$ $0$'s following the $i$th $1$. E.g. a six-vertex clique would be represented by the sequence $110100100010000100000$. This is the lexicographically smallest sequence that could possibly begin a topological ordering of a labeled DAG given by this construction (replacing any of the $1$'s by $0$'s would give a sequence with more edges than could be found in a simple graph with that many vertices) and it can only be the beginning of a topological ordering when $G$ contains the desired clique.