  This article is the third in a series of three articles, the aim of which is
to study various correspondences between four enumerative theories associated
to a surface $S$: Gromov-Witten theory of $S^{[n]}$, orbifold Gromov-Witten
theory of $[S^{(n)}]$, relative Gromov-Witten theory of $S \times C$ for a
nodal curve $C$ and relative Donaldson-Thomas theory of $S \times C$. In this
article, we introduce a one-parameter stability condition, termed
$\epsilon$-admissibility, for relative maps from nodal curves to $X \times C$.
If $X$ is a point, $\epsilon$-admissibility interpolates between moduli spaces
of stable maps to $C$ relative to some fixed points and moduli spaces of
admissible covers with arbitrary ramifications over the same fixed points and
simple ramifications elsewhere on $C$. Using Zhou's calibrated tails, we prove
wall-crossing formulas relating invariants for different values of $\epsilon$.
If $X$ is a surface $S$, we use this wall-crossing in conjunction with author's
quasimap wall-crossing to show that relative Pandharipande-Thomas/Gromov-Witten
correspondence of $S\times C$ and Ruan's extended crepant resolution conjecture
of the pair $S^{[n]}$ and $[S^{(n)}]$ are equivalent up to explicit
wall-crossings. We thereby prove crepant resolution conjecture for 3-point
genus-0 invariants in all classes, if $S$ is a toric del Pezzo surface.
