  We study pairs $(f, \Gamma)$ consisting of a non-Archimedean rational
function $f$ and a finite set of vertices $\Gamma$ in the Berkovich projective
line, under a certain stability hypothesis. We prove that stability can always
be attained by enlarging the vertex set $\Gamma$. As a byproduct, we deduce
that meromorphic maps preserving the fibers of a rationally-fibered complex
surface are algebraically stable after a proper modification. The first article
in this series examined the limit of the equilibrium measures for a
degenerating 1-parameter family of rational functions on the Riemann sphere.
Here we construct a convergent countable-state Markov chain that computes the
limit measure.
  A classification of the periodic Fatou components for non-Archimedean
rational functions, due to Rivera-Letelier, plays a key role in the proofs of
our main theorems. The appendix contains a proof of this classification for all
tame rational functions.
