We report the analysis of the three-body ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}B\overline{B}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}$, $B{\overline{B}}^{*}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}$, and ${B}^{*}{\overline{B}}^{*}{\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}$ processes, including the first observations of the ${Z}_{b}^{\ifmmode\pm\else\textpm\fi{}}(10610)\ensuremath{\rightarrow}[B{\overline{B}}^{*}+\mathrm{c}.\mathrm{c}.{]}^{\ifmmode\pm\else\textpm\fi{}}$ and ${Z}_{b}^{\ifmmode\pm\else\textpm\fi{}}(10650)\ensuremath{\rightarrow}[{B}^{*}{\overline{B}}^{*}{]}^{\ifmmode\pm\else\textpm\fi{}}$ transitions that are found to dominate the corresponding final states. We measure Born cross sections for the three-body production of $\ensuremath{\sigma}({e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}[B{\overline{B}}^{*}+\mathrm{c}.\mathrm{c}.{]}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\pi}}^{\ensuremath{\mp}})=[17.4\ifmmode\pm\else\textpm\fi{}1.6(\text{stat})\ifmmode\pm\else\textpm\fi{}1.9(\text{syst})]\text{ }\text{ }\mathrm{pb}$ and $\ensuremath{\sigma}({e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}[{B}^{*}{\overline{B}}^{*}{]}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\pi}}^{\ensuremath{\mp}})=[8.75\ifmmode\pm\else\textpm\fi{}1.15(\text{stat})\ifmmode\pm\else\textpm\fi{}1.04(\text{syst})]\text{ }\text{ }\mathrm{pb}$ and set a 90% C.L. upper limit of $\ensuremath{\sigma}({e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}[B\overline{B}{]}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\pi}}^{\ensuremath{\mp}})<2.9\text{ }\text{ }\mathrm{pb}$. The results are based on a $121.4\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ data sample collected with the Belle detector at a center-of-mass energy near the $\mathrm{\ensuremath{\Upsilon}}(10860)$ peak.