  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
finite-size but large rings through the use of a recently developed Pfaffian
method [N. Wu, Phys. Rev. E 101, 042108 (2020)]. Time dependence of the
transverse and longitudinal magnetizations, single-spin purity, expectation
value of the string operator $X_j=\prod^{j-1}_{l=1}\sigma^z_l\sigma^x_j$,
several equal-time two-site correlators, and pairwise concurrence after
quenches to different phases are numerically studied. Our main findings are
that (i) The expectation value of a generic odd operator approaches zero in the
long-time limit; (ii) $\langle X_j\rangle_t$ exhibits $j$-independent
exponential decay for a quench to $g=1$ and the time at which $\langle
X_j\rangle_t$ reaches its first maximum scales linearly with $j$; (iii) 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 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; (iv) The nearest-neighbor entanglement
acquires a finite plateau value that increases with increasing $g$, and
approaches a saturated value $\sim0.125$ for $g\gg1$.
