  This is the second of two papers in which we investigate the properties of
displacement functions of automorphisms of free groups (more generally, free
products) on the Culler-Vogtmann Outer space $CV_n$ and its simplicial
bordification. We develop a theory for both reducible and irreducible
autormorphisms. As we reach the bordification of $CV_n$ we have to deal with
general deformation spaces, for this reason we developed the theory in such
generality. In first paper~\cite{FMpartI} we studied general properties of the
displacement functions, such as well-orderability of the spectrum and the
topological characterization of min-points via partial train tracks (possibly
at infinity). This paper is devoted to proving that for any automorphism
(reducible or not) any level set of the displacement function is connected. As
an application, this result provides a stopping procedure for brute force
search algorithms in $CV_n$. We use this to reprove two known algorithmic
results: the conjugacy problem for irreducible automorphisms and detecting
irreducibility of automorphisms. Note: the two papers were originally packed
together in the preprint arxiv:1703.09945. We decided to split that paper
following the recommendations of a referee.
