Near-threshold collisional dynamics in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>e</mml:mi><mml:mo>−</mml:mo></mml:msup><mml:msup><mml:mi>e</mml:mi><mml:mo>+</mml:mo></mml:msup><mml:mi>p</mml:mi></mml:mrow></mml:math> system

We study ${e}^{+}\ensuremath{-}\mathrm{H}(n$) and $\mathrm{Ps}(n)\ensuremath{-}p$ collisions near the three-body breakup threshold and thresholds for the charge-transfer processes. We show that classical trajectory Monte Carlo (CTMC) simulations for the three-body breakup agree reasonably well in this energy region with quantum-mechanical convergent close-coupling (CCC) calculations even if the initial hydrogen atom or positronium atom is in the ground state. The threshold behavior of the three-body breakup cross section in ${e}^{+}$-H($1s$) and $\mathrm{Ps}(1s)\ensuremath{-}p$ collisions agrees with the Wannier law with Klar's exponent and obeys the classical scaling laws, although some deviation from the Klar-Wannier behavior is observed in the CCC results. Below the threshold the agreement between CTMC and CCC disappears. In particular the CTMC method fails completely for the processes of H formation in $\mathrm{Ps}(1s)\ensuremath{-}p$ collisions and Ps formation in ${e}^{+}$-H collisions well below the three-body breakup threshold. For higher initial states the CTMC results below the threshold improve substantially, in accordance with the correspondence principle. This is explained by comparing the quantum-mechanical threshold laws with the classical laws.