  A locally surjective homomorphism from a graph $G$ to a graph $H$ is an
edge-preserving mapping from $V(G)$ to $V(H)$ that is surjective in the
neighborhood of each vertex in $G$. In the list locally surjective homomorphism
problem, denoted by LLSHom($H$), the graph $H$ is fixed and the instance
consists of a graph $G$ whose every vertex is equipped with a subset of $V(H)$,
called list. We ask for the existence of a locally surjective homomorphism from
$G$ to $H$, where every vertex of $G$ is mapped to a vertex from its list. In
this paper, we study the complexity of the LLSHom($H$) problem in $F$-free
graphs, i.e., graphs that exclude a fixed graph $F$ as an induced subgraph. We
aim to understand for which pairs $(H,F)$ the problem can be solved in
subexponential time.
  We show that for all graphs $H$, for which the problem is NP-hard in general
graphs, it cannot be solved in subexponential time in $F$-free graphs unless
$F$ is a bounded-degree forest or the ETH fails. The initial study reveals that
a natural subfamily of bounded-degree forests $F$ that might lead to some
tractability results is the family $\mathcal S$ consisting of forests whose
every component has at most three leaves. In this case, we exhibit the
following dichotomy theorem: besides the cases that are polynomial-time
solvable in general graphs, the graphs $H \in \{P_3,C_4\}$ are the only
connected ones that allow for a subexponential-time algorithm in $F$-free
graphs for every $F \in \mathcal S$ (unless the ETH fails).
