  We study the non-equilibrium dynamics of a homogeneous quantum Ising ring
after a quench, in which the transverse field $g$ suddenly changes from zero to
a nonzero value. The long-timescale reduced dynamics of a single spin and of
two nearest-neighbor spins, which involves the evaluation of expectation values
of odd operators that break the fermion parity, is exactly obtained for
\emph{finite-size} but large rings through the use of a recently developed
Pfaffian method [Phys. Rev. E 101, 042108 (2020)]. The time dependence of the
transverse and longitudinal magnetizations, single-spin purity, several
equal-time two-site correlators, and pairwise concurrence after quenches to
different phases are numerically studied. Our main findings are that (i) For
quenches to $g\geq1$, the expectation value of any generic odd operator
approaches zero in the long-time limit. In particular, the string operator
$\prod^{j-1}_{l=1}\sigma^z_l\sigma^x_j$ exhibits $j$-independent exponential
decay for a quench to $g=1$; (ii) The single-spin purity dynamics is mainly
controlled by $\langle\sigma^x_j\rangle_t$ ($\langle\sigma^z_j\rangle_t$) for a
quench to $g<1$ ($g\geq 1$). For quenches to the disordered phase with $g\gg1$,
the single-spin tends to be in the maximally mixed state with
$\langle\sigma^\alpha_j\rangle_t\approx0$, and the transverse and longitudinal
correlators $\langle\sigma^z_j\sigma^z_{j+1}\rangle_t$ and
$\langle\sigma^x_j\sigma^x_{j+1}\rangle_t$ respectively approaches $-0.25$ and
$0.5$ in the thermodynamic limit; (iii) The nearest-neighbor entanglement
acquires a finite plateau value that increases with increasing $g$, and
approaches a saturated value $\sim0.125$ for $g\gg1$.
