  We report on measurements of branching fractions ($\mathcal{B}$) and
CP-violating charge asymmetries ($\mathcal{A}_{\rm CP}$) of multibody charmless
$B$ decays reconstructed by the Belle II experiment at the SuperKEKB
electron-positron collider. We use a sample of collisions collected in 2019 and
2020 at the $\Upsilon(4S)$ resonance and corresponding to $62.8$ fb$^{-1}$ of
integrated luminosity. We use simulation to determine optimized event
selections. The $\Delta E$ and $M_{\rm bc}$ distributions of the resulting
samples are fit to determine signal yields of approximately 690, 840, and 380
decays for the channels $B^+ \to K^+K^-K^+$, $B^+ \to K^+\pi^-\pi^+$, and $B^0
\to K^+\pi^-\pi^0$, respectively. These yields are corrected for efficiencies
determined from simulation and control data samples to obtain $\mathcal{B}(B^+
\to K^+K^-K^+) = [35.8 \pm 1.6(\rm stat) \pm 1.4 (\rm syst)]\times 10^{-6}$,
$\mathcal{B}(B^+ \to K^+\pi^-\pi^+) = [67.0 \pm 3.3 (\rm stat)\pm 2.3 (\rm
syst)]\times 10^{-6}$, $\mathcal{B}(B^0 \to K^+\pi^-\pi^0) = [38.1 \pm 3.5 (\rm
stat)\pm 3.9 (\rm syst)]\times 10^{-6}$, $\mathcal{A}_{\rm CP}(B^+ \to
K^+K^-K^+) = -0.103 \pm 0.042(\rm stat) \pm 0.020 (\rm syst)$,
$\mathcal{A}_{\rm CP}(B^+ \to K^+\pi^-\pi^+) = -0.010 \pm 0.050 (\rm stat)\pm
0.021(\rm syst)$, and $\mathcal{A}_{\rm CP}(B^0 \to K^+\pi^-\pi^0) = 0.207 \pm
0.088 (\rm stat)\pm 0.011(\rm syst)$. Results are consistent with previous
measurements and demonstrate detector performance comparable with the best
Belle results.
