We study pairs (f, Γ) consisting of a non-Archimedean rational function f and a finite set of vertices Γ in the Berkovich projective line, under a certain stability hypothesis. We prove that stability can always be attained by enlarging the vertex set Γ. 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.
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